vector assignment class 11

Vectors Class 11 Physics | Notes

All physical quantities can be broadly classified into two categories: vector quantity and scalar quantity.

Vector Quantity:

A physical quantity which has both magnitude and direction and obeys the rules of vector algebra is known as vector or vector quantity. It is denoted by alphabetical letter(s) with an arrow- head over it.

Electric current and pressure have both magnitude and direction but they do not obey the rules of vector algebra. So, electric current and pressure are scalar quantities (not vector). 

For Example: displacement, velocity, acceleration, force etc.

Scalar Quantity:

A physical quantity which has magnitude only but no direction is called a scalar quantity. Scalar quantity obeys the rules of simple algebra (scalar algebra). It is denoted by an alphabetical letter.

For Example: mass, length, time, distance, speed etc.

Representation of a vector:

Any vector ($\overrightarrow{a}$) can be expressed as

$\overrightarrow{a}$ = a 1 $\widehat{i}$ + a 2 $\widehat{j}$ + a 3 $\widehat{k}$

Where a 1 , a 2 and a 3 are components of $\overrightarrow{a}$ along X, Y and Z-axis respectively and $\widehat{i}$, $\widehat{j}$ and $\widehat{k}$ are unit vectors along X, Y and Z-axis respectively.

Modulus/Magnitude of a vector:

The modulus of any vector $\overrightarrow{a}$ = x$\widehat{i}$ + y$\widehat{j}$ + z$\widehat{k}$ is denoted by $\left| \overrightarrow{a} \right|$ and given by:

$\left| \overrightarrow{a} \right|$ = $\sqrt{{{x}^{2}}+{{y}^{2}}+{{z}^{2}}}$ unit.

Q. Find the magnitude (modulus) of given vector $\overrightarrow{a}$ = 3$\widehat{i}$ – 2$\widehat{j}$ + $\widehat{k}$.

Solution: The given vector is $\overrightarrow{a}$ = 3$\widehat{i}$ – 2$\widehat{j}$ + $\widehat{k}$.

Then modulus of $\overrightarrow{a}$ is

$\left| \overrightarrow{a} \right|$ = $\sqrt{{{3}^{2}}+{{(-2)}^{2}}+{{1}^{2}}}$

= $\sqrt{9+4+1}$

= $\sqrt{14}$ unit.

Types of vector:

1. unit vector:.

A vector which has magnitude one (unity) is called unit vector. It is denoted by an alphabetical letter with the cap over it.

 Example:   $\widehat{a}$, $\widehat{b}$, $\widehat{i}$, $\widehat{j}$ ,$\widehat{k}$ etc. Mathematically, the unit vector along a vector, $\overrightarrow{a}$ is given by

$\widehat{a}$ = $\frac{\overrightarrow{a}}{\left| \overrightarrow{a} \right|}$

 $\widehat{i}$, $\widehat{j}$ and $\widehat{k}$ are the unit vectors along x, y and z – axis respectively.                                               

Q. Find the unit vector along (of) $\overrightarrow{a}$ = 2$\widehat{i}$ – $\widehat{j}$ + 2$\widehat{k}$.

Solution: The given vector is $\overrightarrow{a}$ = 2$\widehat{i}$ – $\widehat{j}$ + 2$\widehat{k}$.

The modulus of$\overrightarrow{a}$ is

$\left| \overrightarrow{a} \right|$ = $\sqrt{{{\mathbf{2}}^{2}}+{{(-\mathbf{1})}^{2}}+{{\mathbf{2}}^{2}}}$

= $\sqrt{\mathbf{4}+\mathbf{1}+\mathbf{4}}$ = $\sqrt{\mathbf{9}}$ = 3 unit.

Now, the unit vector along $\overrightarrow{a}$ is given by

$\text{ }$= $\frac{2\widehat{i}-\widehat{j}+2\widehat{k}}{3}$

$\text{ }$= $\frac{2}{3}$$\widehat{i}$ –$\frac{1}{3}$ $\widehat{j}$ + $\frac{2}{3}$$\widehat{k}$.

A Quick Question For You: Find the magnitude of $\overrightarrow{b}$= $\frac{2}{3}$$\widehat{i}$ –$\frac{1}{3}$ $\widehat{j}$ + $\frac{2}{3}$$\widehat{k}$. [Ans: 1]

2. Null vector:

A vector which has magnitude zero is called null vector. This vector is generally used to indicate the direction of physical quantity. i.e $\overrightarrow{a}$ is null vector if $\left| \overrightarrow{a} \right|$ = 0

3. Parallel vector:

Two vectors of the same or different magnitude are said to be parallel vectors if they act in the same direction.

In figure, $\overrightarrow{A}$ and $\overrightarrow{B}$ are parallel vectors.

4. Anti-parallel vectors (Unlike Parallel):

Two vectors of same and opposite magnitude are said to be anti-parallel vectors if they act in the opposite direction.

In figure, $\overrightarrow{A}$ and $\overrightarrow{B}$ are anti-parallel vectors.

5. Equal vectors:

Two vectors are said to be equal vectors if they have equal magnitude and act in the same direction.

In figure, $\overrightarrow{A}$ and $\overrightarrow{B}$  are equal vectors.

6. Negative vector:

The negative vectors of $\overrightarrow{A}$ is defined as a vector which has equal magnitude and opposite direction to that of $\overrightarrow{A}$.

In figure, $\overrightarrow{A}$ and $\overrightarrow{B}$ are negative vectors of each other. 

7. Collinear vectors:

Two or more vectors are said to be collinear if they act along the same straight line. 

In figure, $\overrightarrow{A}$ , $\overrightarrow{B}$ and $\overrightarrow{C}$ are collinear vectors.

8. Coplanar vectors:

Two or more vectors are said to be coplanar if they act on a same plane.

In figure, $\overrightarrow{A}$, $\overrightarrow{B}$ and $\overrightarrow{C}$ are coplanar vectors.

9. Position Vectors:

A vector whose initial point is origin and terminal point be any point ‘P’

In figure, $\overrightarrow{OA}$ = $\overrightarrow{r}$ be the position vector of point ‘P’.

10. Co-initial vectors:

Two or more vectors are said to be co- initial if they have the same starting point.

In figure, $\overrightarrow{A}$ and $\overrightarrow{B}$ are co-initial vectors.

11. Co- terminal vectors:

Two or more vectors are said to be co- terminal if they have the same terminal (final) point.

In figure, $\overrightarrow{A}$ , $\overrightarrow{B}$ and $\overrightarrow{C}$ co- terminal vectors. 

12. Orthogonal vectors:

Two vectors are said to be orthogonal if they are perpendicular to each other.

In figure, $\overrightarrow{A}$ and $\overrightarrow{B}$ are orthogonal vectors. 

Composition of vectors:

The process of obtaining a single vector from two or more vectors is called the composition of vectors. 

The single vector obtained as a result is called the resultant vector .

Triangle Law of vector addition:

Statement: When two vectors acting simultaneously at a point be represented by two sides of a triangle taken in same order both in magnitude and direction then the third side taken in reverse order represents their resultant both in magnitude and direction.

Let us consider two vectors acting simultaneously at a point be represented by two sides AB and BC of triangle ABC taken in same order both in magnitude and direction.

According to triangle law of vector addition, third side AC taken in reverse order represents their resultant both in magnitude and direction i.e. $\overrightarrow{AC}$ = $\overrightarrow{R}$ = $\overrightarrow{P}$ + $\overrightarrow{Q}$. Let us produce AB to N and draw CN perpendicular to it (AN). Let the angle between two vectors $\overrightarrow{P}$ and $\overrightarrow{Q}$ be

$\angle $CBN=$\theta $ and

$\alpha $ = $\angle $CAB be the direction of resultant vector with first vector $\overrightarrow{P}$.

In right angled $\Delta $ BNC,

sin$\theta $ = $\frac{CN}{BC}$ =$\frac{CN}{Q}$ $\therefore $CN = Qsin$\theta $

& cos$\theta $ = $\frac{BN}{BC}$ = $\frac{BN}{Q}$ $\therefore $BN = Qcos$\theta $

Now, in right angled $\Delta $ ANC,

h 2 = p 2 + b 2

Or, (AC) 2 = (CN) 2 + (AN) 2

Or, R 2 = (Qsin$\theta $) 2 + (AB+ BN) 2

Or, R 2 = Q 2 sin$\theta $ 2 + (P+ Qcos$\theta $) 2

Or, R 2 = Q 2 sin$\theta $ 2 + P 2 +2PQcos$\theta $ + Q 2 cos$\theta $ 2

Or, R 2 = P 2 +2PQcos$\theta $ +Q 2 (sin 2 $\theta $ + cos 2 $\theta $)

Or, R 2 = P 2 +2PQcos$\theta $ +Q 2

Or, R = $\sqrt{{{P}^{2}}+2PQcos\theta +{{Q}^{2}}}$

which is the required expression for magnitude of resultant vector.

Again in right angled $\Delta $ANC

tan$\alpha $ = $\frac{CN}{AN}$

or, tan$\alpha $ = $\frac{Qsin\theta }{P+Qcos\theta }$

or, $\alpha $ = tan -1 ($\frac{Qsin\theta }{P+Qcos\theta }$), which is the required direction of resultant vector $\overrightarrow{R}$ with first vector $\overrightarrow{P}$.

Parallelogram law of vector addition:

Statement: When two vectors acting simultaneously at a point be represented by two adjacent sides of a parallelogram starting from the same point both in magnitude and direction then the diagonal starting from the same point represents their resultant both in magnitude and direction.

Let us consider two vectors $\overrightarrow{P}$ and $\overrightarrow{Q}$ acting simultaneously at a point be represented by two adjacent sides AB and AD starting from the same point of a parallelogram ABCD both in magnitude and direction. According to the parallelogram law of vector addition, the diagonal AC starting from the same point represents their resultant both in magnitude and direction.

i.e. $\overrightarrow{AC}$ = $\overrightarrow{R}$ = $\overrightarrow{P}$ + $\overrightarrow{Q}$. Let us produce AB to N and draw CN perpendicular to it (AN). Let the angle between two vectors $\overrightarrow{P}$ and $\overrightarrow{Q}$ be $\angle $BAD = $\theta $ and

$\overrightarrow{BC}$ = $\overrightarrow{AD}$ = $\overrightarrow{Q}$ [being the opposite sides of parallelogram ABCD]

$\angle $CBN = $\angle $BAD = $\theta $ [being corresponding angles between parallel lines BC and AD]

Special cases:

(i) When two vectors act in the same direction i.e. $\theta $=0°

This is the condition of maximum value of magnitude of $\overrightarrow{R}$.

i.e. R = P + Q

The direction of the resultant vector is same as that of $\overrightarrow{P}$ and $\overrightarrow{Q}$.

(ii) when two vectors act in the opposite direction i.e.d$\theta $ =180°

then R = $\sqrt{{{P}^{2}}+2PQcos\theta +{{Q}^{2}}}$

R = $\sqrt{{{P}^{2}}+2PQcos{{180}^{\circ }}+{{Q}^{2}}}$

R = $\left| P-Q \right|$

This is the condition of minimum value of magnitude of $\overrightarrow{R}$. The direction of the resultant is same as that of larger vector.

(iii) When two vectors act perpendicular to each other i.e. $\theta $ = 90°

R = $\sqrt{{{P}^{2}}+2PQcos{{90}^{\circ }}+{{Q}^{2}}}$

R = $\sqrt{{{P}^{2}}+{{Q}^{2}}}$ is the magnitude of $\overrightarrow{R}$

The direction of resultant $\overrightarrow{R}$ is in between $\overrightarrow{P}$ and $\overrightarrow{Q}$, slanted towards the larger vector.

(iv) When two vectors have equal magnitude. $\left| \overrightarrow{P} \right|$ = $\left| \overrightarrow{Q} \right|$ = x

R = $\sqrt{{{x}^{2}}+2xxcos\theta +{{x}^{2}}}$

R = $\sqrt{2{{x}^{2}}+2{{x}^{2}}cos\theta }$

R = $\sqrt{2{{x}^{2}}(1+\cos \theta )}$

R = 2x cos$\frac{\theta }{2}$ is the magnitude of $\overrightarrow{R}$. The direction of $\overrightarrow{R}$ is exactly in between $\overrightarrow{P}$ and $\overrightarrow{Q}$. i.e. $\alpha $ = $\frac{\theta }{2}$

Polygon law of vector addition:

It states that when a number of vector acting simultaneously at a point be represented by different sides of a polygon taken in same order both in magnitude and direction then, the closing side of polygon taken in reverse order represents the resultant both in magnitude and direction.

Let us consider five vectors $\overrightarrow{A}$, $\overrightarrow{B}$, $\overrightarrow{C}$, $\overrightarrow{D}$ and $\overrightarrow{E}$ acting simultaneously at a point be represented by different (n-1) sides of a polygon OA, AB, BC, CD, DE taken in same order both in magnitude and direction then the closing side (n th ) of the polygon taken in reverse order represents their resultant both in magnitude and direction. 

i.e. $\overrightarrow{OA}$ + $\overrightarrow{AB}$ + $\overrightarrow{BC}$ + $\overrightarrow{CD}$ + $\overrightarrow{DE}$ = $\overrightarrow{OE}$

$\overrightarrow{A}$ + $\overrightarrow{B}$ + $\overrightarrow{C}$ + $\overrightarrow{D}$ + $\overrightarrow{E}$ = $\overrightarrow{R}$

Subtraction of vectors: When $\overrightarrow{B}$ is subtracted from $\overrightarrow{A}$ then using parallelogram law of vector addition the resultant vector $\overrightarrow{R}$ i.e. ($\overrightarrow{A}$ – $\overrightarrow{B}$) can be obtained as

Resolution of a vector:

The process of splitting a vector into its constituent components is called resolution of a vector.

If resolution of a vector is carried out in such a way that the two components are perpendicular to each other then the components are known as rectangular components of the vector .

Let the vector $\overrightarrow{A}$ be represented by the side OA both in magnitude and direction.

Let the vector $\overrightarrow{A}$ has A x and A y components along horizontal and vertical direction such that OB = A x and OC= A y

Let ‘$\theta $’ be the direction of $\overrightarrow{A}$ with positive x-axis.

Let us complete a parallelogram OBAC as shown in figure.

From figure,

BA = OC = A y

X-component of $\overrightarrow{A}$, A x = Acos$\theta $……..(i)

Y-component of $\overrightarrow{A}$, A y = Asin$\theta $……..(ii)

Squaring and adding eqn (i) and (ii)

A x 2 + A y 2 = A 2 [(cos$\theta $) 2 +(sin$\theta $) 2 ]

A x 2 + A y 2 = A 2

A = $\sqrt{\mathbf{A}_{x}^{2}+\mathbf{A}_{y}^{2}}$ is magnitude of $\overrightarrow{A}$ in terms of its rectangular components.

Similarly, tan$\theta $ = $\frac{AB}{OB}$ = $\frac{{{A}_{y}}}{{{A}_{x}}}$

$\theta $ = tan -1 ($\frac{{{A}_{y}}}{{{A}_{x}}}$) is the direction of $\overrightarrow{A}$ in terms of its rectangular components.

Multiplication of a vector

(i) multiplication of a vector by a number:.

When $\overrightarrow{a}$ is multiplied by a number (n) the result n$\overrightarrow{a}$ is a vector quantity whose magnitude is ‘n’ times that of $\overrightarrow{a}$ and direction is same as that of $\overrightarrow{a}$.

(ii) Multiplication of a vector by a scalar:

When $\overrightarrow{a}$ is multiplied by a scalar ‘m’ (say) the result, m$\overrightarrow{a}$ is a vector quantity. In this case the product obtained is a new physical quantity which has the same direction as that of $\overrightarrow{a}$.

For example, m$\overrightarrow{a}$=$\overrightarrow{F}$ where m is mass, $\overrightarrow{a}$ is the acceleration and $\overrightarrow{F}$is force.

(iii) Multiplication of a vector by another vector:

A. dot product (scalar product):.

The dot product of two vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ is denoted by$\overrightarrow{a}$.$\overrightarrow{b}$ and given by$\overrightarrow{a}$.$\overrightarrow{b}$ = a b cos$\theta $

where ‘$\theta $’ be the angle between two vectors $\overrightarrow{a}$ and $\overrightarrow{a}$.

The dot product of two vectors in a scalar quantity that is why it is also called a scalar product.

Also, if $\overrightarrow{a}$ = a 1 $\widehat{i}$ + a 2 $\widehat{j}$ + a 3 $\widehat{k}$ and $\overrightarrow{b}$ = b 1 $\widehat{i}$ + b 2 $\widehat{j}$ + b 3 $\widehat{k}$

then$\overrightarrow{a}$.$\overrightarrow{b}$ = a 1 b 1 + a 2 b 2 + a 3 b 3

Properties of scalar / dot product.

1. The dot product of $\overrightarrow{a}$ and $\overrightarrow{b}$ is maximum when $\theta $ = 0°

i.e. $\overrightarrow{a}$.$\overrightarrow{b}$ = a b cos0°= ab (Maximum)

2. A dot product of a and b is zero when if $\theta $ = 90°

i.e. $\overrightarrow{a}$.$\overrightarrow{b}$ = a b cos90°= 0

3. The scalar / dot product of two vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ follows commutative law$\overrightarrow{a}$.$\overrightarrow{b}$ = $\overrightarrow{b}$.$\overrightarrow{a}$

4. The dot product of two vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ follows distributive law. i.e. $\overrightarrow{a}$.($\overrightarrow{b}$+$\overrightarrow{c}$) =$\overrightarrow{a}$.$\overrightarrow{b}$ +$\overrightarrow{a}$.$\overrightarrow{c}$

5. The square of a vector is a scalar quantity.

i.e. ($\overrightarrow{a}$) 2 =$\overrightarrow{a}$.$\overrightarrow{a}$ = a a cos0°= a 2

$\therefore $ ($\overrightarrow{a}$) 2 = a 2 is a scalar quantity.

Scalar / dot product of unit vectors.

(i) $\widehat{i}$ . $\widehat{i}$ = $\left| \widehat{i} \right|\text{ }\left| \widehat{i} \right|$cos0°= 1 i.e. $\widehat{i}$ . $\widehat{i}$ = $\widehat{j}$.$\widehat{j}$= $\widehat{j}$ + $\widehat{k}$. $\widehat{k}$ = 1

(ii) $\widehat{i}$. $\widehat{j}$= $\left| \widehat{i} \right|\text{ }\left| \widehat{j} \right|$cos90°= 0 i.e. $\widehat{i}$ . $\widehat{j}$= $\widehat{j}$.$\widehat{k}$+ $\widehat{k}$. $\widehat{i}$ = 0

b. Cross product (Vector product) of two vectors:

The vector (cross) product of two vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ is denoted by $\overrightarrow{a}$×$\overrightarrow{b}$ and given by

$\overrightarrow{a}$×$\overrightarrow{b}$ = $\left| \overrightarrow{a} \right|$ $\left| \overrightarrow{b} \right|$ sin$\theta $ $\widehat{n}$

= a b sin$\theta $ $\widehat{n}$

where, a b sin$\theta $ is the magnitude of $\overrightarrow{a}$×$\overrightarrow{b}$ and and ‘$\widehat{n}$’ is its direction which is perpendicular to both $\overrightarrow{a}$ and $\overrightarrow{b}$.

The direction of $\overrightarrow{a}$×$\overrightarrow{b}$ is obtained by right hand thumb rule or right hand screw rule.

The vector $\overrightarrow{a}$×$\overrightarrow{b}$ is perpendicular to both $\overrightarrow{a}$ and $\overrightarrow{b}$. Since, the cross product of $\overrightarrow{a}$ and $\overrightarrow{b}$ is again a vector so, it is also called vector product.

if $\overrightarrow{a}$ = a 1 $\widehat{i}$ + a 2 $\widehat{j}$ + a 3 $\widehat{k}$ and $\overrightarrow{a}$ = b 1 $\widehat{i}$ + b 2 $\widehat{j}$ + b 3 $\widehat{k}$

$\overrightarrow{a}$ = (a 2 b 3 – a 3 b 2 ) $\widehat{i}$ +(a 3 b 1 – a 1 b 3 ) $\widehat{j}$ + (a 1 b 2 – a 2 b 1 )$\widehat{k}$

Properties of vector / cross product:

• The magnitude of vector product is maximum if $\theta $ = 90°

    i.e. $\left| \overrightarrow{a}\times \overrightarrow{b} \right|$ = a b sin90°= ab (max m )

• Similarly,  the magnitude of vector product is zero if $\theta $ = 0°
• i.e. $\left| \overrightarrow{a}\times \overrightarrow{b} \right|$ = a b sin0°= 0 (zero)
• The vector product does not obey commutative law.

     i.e. $\overrightarrow{a}\times \overrightarrow{b}\ne \overrightarrow{b}\times \overrightarrow{a}$ but $\vec{a}\times \vec{b}=-(\vec{b}\times \vec{a})$

     4. The vector product always obeys distributive law.

     i.e. $\overrightarrow{a}\times (\overrightarrow{b}+\overrightarrow{c}\text{)=}\overrightarrow{a}\times \overrightarrow{b}+\overrightarrow{a}\times \overrightarrow{c}$

     5. The magnitude of cross product of a vector with itself is zero i.e.

$\left| \overrightarrow{a}\times \overrightarrow{a} \right|$= a a sin0° = 0 (zero)

Physical Significance (geometrical meaning) of vector product: 

If two vector $\overrightarrow{a}$ and $\overrightarrow{b}$ are represented by two adjacent sides of a parallelogram both in magnitude and direction then the magnitude of $\overrightarrow{a}$×$\overrightarrow{b}$ gives the area of that parallelogram.

i.e. $\left| \overrightarrow{a}\times \overrightarrow{b} \right|$ = ab sin$\theta $ = area of parallelogram OACB.

Some important numerical problems of Vector:

Q.1 A disoriented physics professor drives 3.25 km north, then 4.75 km west and then 1.50 km south. Find the magnitude and direction of the resultant displacement.

Ans: 5.06km, 20.22 o north of west (or 69.78 o W of N).

Q.2 A spelunker is surveying a cave. She follows a passage of 180m straight west, then 210m in a direction 45 o east of south, and then 280m at 30 o east of north. After a fourth unmeasured displacement she finds herself back where she started. Use the method of components to determine the magnitude and direction of the fourth displacement.

Ans: 144m, 41 o S of W.

Q. 3 A cave explorer is surveying a cave. He follows a passage 100m straight east, then 50m in a direction 30 o west of north, then 150m at 45 o west of south. After a fourth unmeasured displacement he finds himself back where he started. Using a scale drawing to determine the forth displacement (magnitude and direction).

Ans: 70.02m in the direction 26.34 o east of north.

Q. 4 A rocket fires two engines simultaneously. One produces a thrust of 725N directly forward, while the other gives a 513N thrust at 32.4 o above the forward direction. Find the magnitude and direction of the resultant force that these engines exert on the rocket.

Ans: 1119N and direction 13.4 o .

Also Read: Kinematics: Notes Class 11 | Physics (Part-I)

4 thoughts on “Vectors Class 11 Physics | Notes”

Thank you so much.its helps me a lot.

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High School Physics

Vectors class 11 Physics revision notes – chapter 4

Last updated on July 5th, 2023 at 08:49 pm

This post covers Vectors class 11 Physics revision notes – chapter 4 with concepts, formulas, applications, numerical, and Questions. These revision notes are good for CBSE, ISC, UPSC, and other exams. This covers the grade 12 Vector Physics syllabus of some international boards as well. Here we have covered Vector fundamentals & types, Laws for vector addition, dot and cross products, & relative velocity of rain with respect to a moving man (with numerical).

There are physical quantities that require both magnitude and direction for their complete description. A simple example of a vector is velocity . The statement that the velocity of a train is 100 Km/hour does not make much sense unless we also tell the direction in which the train is moving.

Force is another such quantity. We must specify not only the magnitude of the force but also the direction in which the force is applied. Such quantities are called vectors. A vector quantity has both magnitude and direction. Some examples of vector quantities in mechanics are displacement, acceleration, momentum, angular momentum, torque, etc.

A vector is represented by a line with an arrow indicating its direction. Take vector AB in Fig. 1. The length of the line represents its magnitude on some scale. The arrow indicates its direction.

[ All our posts on Vector are clubbed here: Vector Physics tutorials . Read these for a better understanding ]

Types of Vectors class 11

Equal vector, negative of a vector, null vector or zero vector, unit vector class 11, addition of vectors class 11, triangle law of vector addition class 11, parallelogram law of vector addition class 11, vector numerical problems class 11 (solve using parallelogram law & triangle law), product of vectors class 11, dot product, vector product, vector numerical problems class 11 (on dot product and cross product), relative velocity of rain to a moving man | vector subtraction numerical class 11.

Two vectors are said to be equal if their magnitudes are equal and they point in the same direction. Three vectors A, B, and C shown in the Figure are equal . We say A = B = C . But D is not equal to A .

A vector (here D in the figure above) that has the same magnitude as A but has opposite direction, is called negative of A , or – A . Thus, D = – A

A vector is said to be a zero or null vector if the magnitude of the vector is zero, i.e., the starting point and the endpoint of the Vector are the same.

A vector is said to be a Unit vector if its magnitude is 1 unit and it has a specified direction. A unit vector is a dimensionless vector having a magnitude of exactly 1. Unit vectors are used to specify a given direction and have no other physical significance. They are used solely as a convenience in describing a direction in space.

It has neither units nor dimensions.

As an example, we can write vector A as A n where a cap(^) on n denotes a unit vector in the direction of A . A unit vector has been introduced to take care of the direction of the vector; the magnitude has been taken care of by A.

The unit vectors along coordinate axes are of particular importance. We shall use the symbols i, j, and k to represent unit vectors pointing in the positive x, y, and z directions, respectively. The “^ hats” on the symbols are a standard notation for unit vectors, but many times just a bold font(like i) is popularly used without using the hat. The unit vectors i, j, and k form a set of mutually perpendicular vectors in a right-handed coordinate system, as shown in Figure. The magnitude of each unit vector equals 1; that is, | i | = | j | = | k | = 1.

The unit vector along the x-axis is denoted by i, along the y-axis by j, and along the z-axis by k. Using this notation, vector A, whose components along the x and y axes are respectively A x and A y , can be written as A = A x i + A y j

Vectors of the same kind only can be added. For example, two forces can be added or two velocities can be added. But a force and a velocity cannot be added. Here we will discuss the Triangle Law and the Parallelogram Law to add vectors.

If two vectors are represented in magnitude and direction by the two sides of a triangle taken in order, the resultant is represented by the third side of the triangle taken in the opposite order. This is called the triangle law of vectors.

The sum of two or more vectors is called the resultant vector. In Fig. above, pr is the resultant of A and B

If a parallelogram can be drawn so that two vectors can be placed with their tails connected as the two adjacent sides of the parallelogram with an angle θ between them, then the diagonal of the parallelogram represents their resultant vector or the vector sum.

Let A and B be the two vectors and let θ be the angle between them as shown in Fig. above. To calculate the vector sum, we complete the parallelogram. Here side PQ represents vector A, side PS represents B and the diagonal PR represents the resultant vector R. Here,α angle is the angle the resultant makes with the base vector and the angle denotes the direction of the resultant or the vector sum.

Resultant vector formulas (Magnitude and direction)

Vector A has a magnitude of 30 and it lies in such a way that it makes an angle of 30 degrees with another vector B of magnitude 40. What is the vector sum or resultant of A and B ?

Solution: Let, R be the vector sum of A and B .

|R| = √[30 2 + 40 2 + 2.30.40 cos 30] = 67.66

If R makes an angle α with A then tanα = (B sinθ)/(A + Bcosθ) here, A = 30, B = 40, θ = 30. tanα = 40 sin 30 / (30 + 40 cos 30) = 20/(30 + 34.64) = .31

α = arctan (.31) = 17.22 degree

Answer: The resultant is 67.66 at 17.22 degrees from A

A and B are two forces where |A| = 3 N and |B| = 4 N.

Can you draw the vector sum of these two forces using the triangle law? Also, find out its magnitude and direction using some suitable formula.

Solution: Using Triangle law, vectors A and B are drawn in the tail to tip way to draw 2 sides of the triangle. R, the 3 rd side of the triangle (in red) shows the vector sum of A and B.

The magnitude of the vector sum can be calculated using the Pythagoras theorem in this case and B is perpendicular to A. So the magnitude of the vector sum: |R| = √(3 2 + 4 2 ) = 5 N

If it makes an angle α with A, then tanα = 4/3

α = arctan (4/3) = 53.13

So, the vector sum is 5 N at 53.13 degrees from A

The next section covers the Multiplication of vectors – dot and cross, formulas, rules, and numerical problems.

The scalar multiplication of two vectors yields a scalar product. Scalar multiplication is also known as the dot product.

The formula of Dot Product

The scalar product or dot product of 2 vectors A and B is expressed by the following equation: A.B = AB cos φ, where φ is the angle between the vectors, A is the magnitude of vector A and B is the magnitude of vector B. The scalar product is also called the dot product because of the dot notation that indicates it.

Rules of Dot product or Scalar product

1) Dot product is a scalar, it is commutative: A.B = B.A = ABcosθ. 2) It is also distributive: A.(B + C) = A.B + A.C .

i . j = j . i = 0 j . k =k . j =0 k . i =i . k =0

i . i = 1 j . j = 1 k . k = 1

4) If, A = (A x i + A y j + A z k ) and B = (B x i + B y j + B z k ) Then find A.B

A . B = (A x i + A y j + A z k ) . (B x i + B y j + B z k ) = (A x i . B x i + A x i . B y j + A x i . B z k ) + (A y j . B x i + A y j . B y j + A y j . B z k ) + (A z k . B x i + A z k . B y j + A z k . B z k )

= (A x B x i . i + A x B y i . j + A x B z i . k ) + (A y B x j . i + A y B y j . j + A y B z j . k ) + (A z B x k . i + A z B y k . j + A z B z k . k ) = A x B x + A y B y + A z B z

We define the vector product to be a vector quantity with a direction perpendicular to this plane (that is, perpendicular to both A and B ) and a magnitude equal to AB sinφ.

Formula of Vector Product or Cross Product

if C = AxB , then C = AB sinφ ……………….. (equation 1) [ Here, A and B are magnitudes of vectors A and B respectively. C is the magnitude of the vector C . And, C = (cross) product of A and B ]

The direction of the product vector C =A × B is given by the right-hand rule. If the right hand is held so that the curling fingers point from A to B through the smaller angle between the two, then the thumb stretched at right angles to fingers will point in the direction of C.

Rules of Vector Product or Cross Product

1) Direction of vector BxA is opposite to that of the vector AxB . This means that the vector product is not commutative .

2) i x j = k j x k =i k x i =j

j x i = – k k x j = – i i x k = – j

i x i = 0 j x j = 0 k x k = 0

3) If, A = (A x i + A y j + A z k ) and B = (B x i + B y j + B z k ) Then find A X B

A X B = (A x i + A y j + A z k ) x (B x i + B y j + B z k ) = (A x i xB x i + A x i x B y j + A x i x B z k ) + (A y j x B x i + A y j x B y j + A y j x B z k ) + (A z k x B x i + A z k x B y j + A z k x B z k )

= (A x B x i x i + A x B y i x j + A x B z i x k ) + (A y B x j x i + A y B y j x j + A y B z j x k ) + (A z B x k x i + A z B y k x j + A z B z k x k ) = 0 + A x B y k + A x B z (- j ) + A y B x (- k ) + 0 + A y B z i + A z B x j + A z B y (- i ) + 0 = (A y B z – A z B y ) i + (A z B x – A x B z ) j + (A x B y – A y B x ) k

1) Find out the dot product of vector A and B where A = 4 i + 5 j + 2k , B = 6 i – 4 j + 3k .

Dot product D = A x B x + A y B y + A z B z = 4.6 + 5(-4) + 2.3 = 24 – 20 + 6 = 10

2) Find out the cross product of vector A and B where A = 4 i + 5 j + 2k , B = 6 i – 4 j + 3k .

Cross Product C = (A y B z – A z B y ) i + (A z B x – A x B z ) j + (A x B y – A y B x ) k

=(5.3 – 2(-4)) i + (2.6 – 4.3) j + (4(-4)-5.6) k = 23 i + 0j -46k = 23i – 46k

In the next section, we have discussed and solved numerical problems related to the Relative velocity of Rain with respect to a Moving Man with the help of Vector subtraction .

Raindrop and moving man – relative velocity (Problem number 1)

Rain is falling vertically at a speed of 35 m/s. A man rides a bicycle with a speed of 12 m/s in the east-to-west direction. What is the direction in which he should hold his umbrella?

Velocity of rain = V r = 35 m/s Velocity of man = V m = 12 m/s east to west

The relative velocity of rain with respect to the man =V = V r – V m

=> V = V r + (– V m ) In the diagram, we have reversed the direction of the man to draw – V m

Now, Let this relative velocity V make an angle θ with the vertical. tan θ = V m / V r = 12/35

θ = arctan (12/35) = 18.9 degrees with the vertical towards the west.

The man has to hold his umbrella at 18.9 degrees with the vertical.

Raindrop and moving man – relative velocity (Problem number 2)

Rain is falling vertically at a speed of x m/s. A man rides a bicycle with a speed of 12 m/s in the east-to-west direction. What is the value of x if the direction in which he holds his umbrella is 21 degrees with the vertical?

Velocity of rain = V r = x m/s

Velocity of man = V m = 12 m/s east to west

=> V= x + (– V m )

In the diagram, we have reversed the direction of the man to draw – V m

If this relative velocity V makes an angle θ = 21 degrees with the vertical. tan 21 = V m / V r = 12/x

=> 0.38 =12/x

x=12/0.38=31.57 m/s

Rain is falling vertically with a speed of 31.57 m/s

Raindrop and moving man – relative velocity (Problem number 3)

Rain is falling vertically at a speed of 40 m/s. A man rides a bicycle with a speed of 12 m/s in the east-to-west direction. What will be the magnitude of the relative velocity of rain with respect to the man? (i.e. What would be the apparent velocity of rain to the cyclist?)

Velocity of rain = V r = 40 m/s

=> V= V r + (– V m )

Now V = √( V r 2 + V m 2 ) = √( 40 2 + 12 2 ) = 41.76 m/s

Hence, the apparent velocity of rain to the cyclist=41.76 m/s

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Class 11 Physics (India)

Course: class 11 physics (india)   >   unit 7.

• Combined vector operations

Vector operations review

What are the basic vector operations, practice set 1: adding and subtracting vectors.

• Your answer should be
• an integer, like 6 ‍  
• a simplified proper fraction, like 3 / 5 ‍  
• a simplified improper fraction, like 7 / 4 ‍  
• a mixed number, like 1   3 / 4 ‍  
• an exact decimal, like 0.75 ‍  
• a multiple of pi, like 12   pi ‍   or 2 / 3   pi ‍  

Practice set 2: Scalar multiplication

Practice set 3: combined operations, want to join the conversation.

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Vector of Class 11

Definition: The physical quantities specified completely by their magnitude as well as direction are called vector quantities. The magnitude and direction alone cannot decide whether a physical quantity is a vector. In addition to the above characteristics, a physical quantity, which is a vector, should follow laws of vector addition. For example, electric current has magnitude as well as direction, but does not follow laws of vector addition. Hence, it is not a vector.

A vector is represented by putting an arrow over it. The length of the line drawn in a convenient scale represents the magnitude of the vector. The direction of the vector quantity is depicted by placing an arrow at the end of the line.

Unit vector

Unit vector is basically used to indicate the direction.

Null vector or zero vector

A vector having zero magnitude and indeterminate direction is called Zero or Null Vector.

Concept of zero vector is helpful in substraction of two equal magnitude vectors in opposite direction and vector product of two parallel vectors.

The concept of null vector is hypothetical but we introduce it only to explain some mathematical results.

Invariancy of the vector

Any vector is invariant so it can be taken anywhere in the space keeping its magnitude and direction same. In other words, the vector remains invariant under translation.

• Laws of Addition of Vectors
• Resolution of Vectors
• Direction cosines
• Addition of vectors by analytical method
• Multiplication of vector by a scalar
• Moment of a force (i.e., Torque) about a point

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current events conversation

What Students Are Saying About Why School Absences Have ‘Exploded’

Chronic absenteeism has increased in American schools since the Covid-19 pandemic. We asked teenagers what they make of the trend.

By The Learning Network

Nationally, an estimated 26 percent of public school students were considered chronically absent last school year, up from 15 percent before the Covid-19 pandemic, according to the most recent data, from 40 states and Washington, D.C., compiled by the conservative-leaning American Enterprise Institute.

The increases have occurred in districts big and small, and across income and race.

In “​ Why School Absences Have ‘Exploded’ Almost Everywhere ,” Sarah Mervosh and Francesca Paris explain:

The trends suggest that something fundamental has shifted in American childhood and the culture of school, in ways that may be long lasting. What was once a deeply ingrained habit — wake up, catch the bus, report to class — is now something far more tenuous. “Our relationship with school became optional,” said Katie Rosanbalm, a psychologist and associate research professor with the Center for Child and Family Policy at Duke University.

In a related Student Opinion question , we asked teenagers if that explanation resonated with them. Had their relationship to school — and school attendance — changed since the pandemic? And if so, what did they make of this shift?

Many students said, yes, school feels different now. Why? They pointed to remote learning changing their routines, an increase in anxiety and a decrease in motivation, the ease of making up schoolwork online and much more. Read their responses in full below.

Thank you to everyone who participated in the conversation on our writing prompts this week, including students from Central Bucks South High School in Warrington, Pa .; Norwood High School in Norwood, Mass.; and West Salem High School in Salem, Ore.

Please note: Student comments have been lightly edited for length, but otherwise appear as they were originally submitted.

Remote learning made students comfortable with missing school.

I believe that there are two main contributors to missing school too much. The first is online school. Myself included. It was very easy to simply leave the call after taking attendance and the teacher wouldn’t realize. Skipping class was easy and you could still get high grades. Transitioning back to real school, kids still held that true. They knew that they could miss school and still do well because covid taught that to them. The second reason is punishment. When you miss school, nothing happens. Class goes on and you have a little extra homework the next day but that’s it. What is the issue with missing class is a very common thought and it’s true. There is very minimal downside to missing school. When I had surgery, I missed a full week of school and within a day and a half, I was fully caught up again. Missing school has just become all too easy.

— Xavier, Pennsylvania

2020 was when our lives completely changed for the worst. We all had to stay inside and stay separate from each other. It was terrible, not being able to talk to my friends, and seeing the death toll on news constantly rise. However, after a year into the pandemic, I believe students realized the power they now had, including me. Now that I am a highschooler, I am going to admit that sometimes I would just mute my class and do whatever I wanted. School became shorter and easier to pass than ever before. That’s why when we all transitioned back into school, it was weird. We all still wanted to get through class the “easy way,” yet now that we were back, it wasn’t possible. This is why we started increasing our absences. The threat of absence has become weak, students are not as afraid to stay out of school. Furthermore the threat of being infected gave just one more reason to be out of school, for the sake of “preventing others from getting sick,” when in reality you feel fine. That is most likely why the absences in school had an exponential increase.

— Joshua, Pennsylvania

Students feel like expectations are lower than they were before the pandemic.

As a student in high school, I’ve come to realize the horrible state our attendance has been in since the pandemic. The reason can be simplified into one idea: laziness. We are lazy, willing to do only enough to get by, no more, no less. If a student doesn’t need to come to a class to obtain the grade they wish to achieve, then they won’t show up. Classes are not challenging enough to make students feel that they are worth going to. My mom is used to getting texts from me during the school day, begging to be excused from a class where “we’re doing nothing” or, “I already finished the work,” which is true, yet I abuse the opportunity to miss class because I know there will be no greater coincidence, I will still be getting an A. Due to my laziness, I would rather be at home taking a nap than sitting in a class with no greater impact on my life.

— Clara, Salem, Oregon

Since the pandemic, schooling has been focused on getting students caught up to where we’re supposed to be. Consequently, more allowances are made for students who don’t do assignments or don’t even show up. And with the switch to all online because of the pandemic, things have never shifted back. If a student misses a day or even a week, they can easily see what they missed and do it and submit it from home. With this option giving them the exact same grade as it would if they actually went to school, it’s no wonder why students are choosing to stay at home or skipping class. Additionally, the pandemic had heightened anxiety levels in students, specifically social anxiety, making them less likely to show up. The allowances made by the school district for students has created a space for students to be lazy and get away with it. This is fostering a negative impact on student work ethic not only now, but also in the future when this generation will be entering the work force.

— Emma, West Salem High School

The period of school shutdowns got students out of their school routines.

When I think back to virtual learning, my brain automatically goes to how stress free it was. I was in sixth grade when Covid first hit and going through a period of my life where I was extremely anxious at school. I believe that this break is exactly what I needed at the time. However, I do believe that in the long run, this online learning time period got a lot of people into the routine of not having a routine. A lot of people at my school would turn their camera off and fall asleep or go on their phones during online learning. I believe that there were times that I did this as well. I also think that this mindset carried through into the grades where I did not have an online/hybrid option. In eighth and ninth grade, I happened to stay home sick, go into school late, or leave early a lot. I think this is due to me not taking school as seriously due to the grading methods that were being used and how some of my teachers were not grading harshly. Now that I am a sophomore in high school, I think I have finally gotten back into the routine of actual schooling and not staying home sick unless I actually feel extremely sick.

— Madison, Pennsylvania

Before the pandemic and as I was growing up, I was the kind of student that wanted perfect attendance. For some odd reason, it made me feel like a better student if I never missed a day. This included turning my parents down when they offered me to go on trips, even though I was only in fourth grade and the work that I would have missed wouldn’t have made an impact in my academic career. However, after the pandemic school began to feel optional. We felt what it was like to fall out of the routine that going to school was and were never able to fully recover from it. I think that having experienced attending school from your bed, in your pajamas has played a major role in the current trend of students receiving more absences. For me, it made me realize that the “0” next to your number of absences didn’t matter as much as I had once thought. As a now highschooler, the school days are long and every class requires an abundance of work and undivided attention that whenever there is a substitute or not much going on, it is easy to decide to leave school. With senior year approaching, everything’s purpose is college and the fact that colleges aren’t able to see how many absences a student has when they apply, does play a role in the increasing number of absences.

— Ava, Miami Country Day School

Because assignments and other materials are online, students find they can keep up with their classes even if they don’t attend school.

Schools have adjusted rules so much that it makes school feel optional. Don’t want to attend class publicly? Take online classes. Don’t want to take “required” state testing? Opt out. Before, school seemed strict, we didn’t have the option to opt out of tests, we didn’t think of taking online school. Yet now, schools make it so easy to skip because everything is simply online. Our assignments, lectures, and teachers are all online. There are no longer requirements in school. What’s the point of attending if we can graduate without taking state testing or attending advisory — also a requirement, yet I no longer have an advisory because my counselors said I don’t need to take it to graduate. It’s confusing. Students have been enabled for over 4 years now since quarantine started. School doesn’t feel mandatory, it’s optional. I’m currently enrolled into 2 AP classes, so I try my best not to miss school. But it’s inevitable, I get sick, I have family situations or maybe I simply don’t feel like attending school. But I see people skip school like nothing. “I didn’t feel like going” is a constant statement I hear. Not many students have the motivation to attend, and simply don’t go because they have a comfort in their head that they can graduate while missing multiple days of school nearly everyday.

— Olivia, Salem, OR

Current absenteeism rates have significantly impacted my learning experience for the past few years. Since the pandemic, there has been a noticeable shift in the perception of the value of education and whether or not attendance is an important factor in a student’s academic success. In the years following 2020, I found myself struggling to make it to class everyday due to my new found efficiency of working at home with my computer. I felt that even if I was not in class personally, I would be able to keep up with my work easily as it was all online regardless. Due to this I would go on trips or skip class purely because I was under the impression that I would be able to continue achieving virtually.

— Ruby, RFHS

Before the pandemic, my attendance was stable but after the pandemic, my absences were piling on. It was difficult to get back in the rhythm of in person school when I had already done a whole year online, but now my attendance in school is definitely getting better. On the other hand, students in my school tend to miss school and it is a rare sight to see a full class. Some students go as far as showing up to class once a week and just do the classwork online. After the pandemic, schools went from paperwork to all online, which is a big reason why students miss all the time, knowing that school work can just be done at home. It has definitely affected students’ grades and goals in life, but hopefully in the future, absences can lower back down.

— Emily, Atrisco Heritage Academy High School

Going to school, and finding the motivation to have as good an attendance record as possible, now feels like more of a struggle.

As students, we’ve developed a comfort in staying in bed during school without having to get ourselves ready to go outside. We had the ability to wake up five minutes before “school” started to get on our zoom calls. Now, we must wake up an hour and a half prior, and make breakfast and pack lunch, before driving to school. The process is tenuous as the article states, but because we’ve accustomed to a different lifestyle, it just makes this one seem like so much more work. I, myself have noticed my difference in attendance after COVID-19. I used to be very obsessed with perfect attendance, but I had 11 absences in my sophomore year, right after coming back from online school. Nowadays, I’m more lenient on myself when it comes to taking a mental health day, because the process can be overwhelming. School is very important, so of course I try to always come in, but sometimes it can be hard. I have not noticed this trend in the world, as well as with myself until this article. It’s enlightening to know that this had not only an effect on me, but all over the country. Hopefully the rates of absenteeism will decrease as time goes on, because we are the future.

— Anisha, New Jersey

Before virtual learning, I never made much of a habit of not turning in work or showing up for class. It was so much easier then but since virtual learning, it had become incredibly difficult for me to focus as well as keep up motivation to continue school. It was easy to skip and nobody really said much about it so it easily became a bad habit. That bad habit eventually leaked into normal school as well and it always sounds so much easier to break out of than it actually is.

— Tayy, NRHS

As the average high school class skipper (only sometimes), in my personal experience, missing out on classes hasn’t really been because of mental health concerns, but more of just lasting laziness from the pandemic. I feel as though I was relatively hard working in middle school/elementary but after a few years off with only half effort assignments, I have grown to become more sluggish and reluctant when it comes to more advanced work while in school. And it makes the option of missing out on classes because of my own reluctance a lot more appealing.

— Luke, Bali, Indonesia

My schedule during the week is get up, get ready for school, go to school, go home, do homework, go to sleep and then I repeat that everyday for 5 days. As much as I don’t want to dread going to school, it’s exhausting having the same schedule repeated everyday of the week. While in school, you have assignments assigned nearly everyday. I feel as though school has had a change in its meaning because of the COVID-19 pandemic. While in quarantine, we were looking at a screen for the whole day and lacked motivation to get assignments done. When we shifted to in person school again, it didn’t change. I now look at school as a task that I need to complete to shape my future. I need to have all my assignments perfect and turned in on time. The meaning of school has turned into a draining task rather than a place that you look forward to going to.

— Jamisan, Salem, Oregon

Some students face challenges in attending class that may have nothing to do with the pandemic.

I don’t believe that students are skipping because it is so easy to catch up and pass, despite their absences. In fact, I know that a lot of people who skip aren’t passing most of their classes. They do this because their parents don’t hold them accountable, and there is always something deeper going on in that student’s life that makes it that much harder for them to find the motivation to go to class. I don’t think making the classes harder will hold students more accountable, but in fact deter them from going to class at all. If a student is aware that they are failing and doesn’t understand the concept of the class, and the class proceeds to become harder, they are going to quickly become unmotivated to go to class in the first place, feeling out of place compared to the other — passing — students in the class. While I don’t have a solution for this problem, myself, I feel that the problem is much broader than we suspect, and the answer will be a much deeper journey to find.

— Kylie, West Salem HS

Schools can do more to get students back in class.

I attend a French school in London and attendance is closely monitored. Absences have to be justified by your parents or you could get into trouble. I think it’s important to attend school as we did before Covid - because as well as learning the curriculum, it is crucial to socialise with your friends and classmates, which is good for your mental health … I wonder if social media could be a factor? If students did not have access to social media or the internet, would they prefer to be in school with their friends? This increase in absenteeism could affect students’ chances of getting into University when they come to finish school or even their opportunities later in life. Students need to be reminded of this more and more perhaps. School helps you to learn not just about facts but also helps to build your emotional quotient & social intelligence — which are all valuable for life.

— Alexandre 14, London

As a current high school junior, my experiences with skipping have been minimal at best, however, I feel strongly that the reason behind skipping is pretty simple. Students don’t care as much about school and the system encourages it. When faced with the choice of sitting in a class and learning about the Patagorian theorem or hanging out with friends, many students are now choosing the latter. The lack of care or effort being put forth in school doesn’t even affect their grades! This is due to certain classes having minimal grades set at 50%, which is 10% away from a pass. This system is actively encouraging people to put minimal effort into a class just to get a pass and graduate. Removing courses like this would certainly raise the importance of getting the work done. Another solution to this problem would be having attendance as a grade, if your grade depends on you being in classes then most would show up. If you have to show up to class to pass then more students would be inclined to do so. The emphasis is on not bending the knee to people who don’t want to show up to class, not giving them a minimal 50%, we should mark attendance for a passing grade, and letting them fail. If we keep letting students skip with minimal consequences then their attitudes won’t change and thus hinder our students’ growth.

— Henry, Salem, OR

Learn more about Current Events Conversation here and find all of our posts in this column .

• Math Article

Vectors, in Maths,  are objects which have both, magnitude and direction. Magnitude defines the size of the vector. It is represented by a line with an arrow, where the length of the line is the magnitude of the vector and the arrow shows the direction. It is also known as Euclidean vector or Geometric vector or Spatial vector or simply “ vector “. 

Two vectors are said to equal if their magnitude and direction are the same. It plays an important role in Mathematics, Physics as well as in Engineering. According to vector algebra, a vector can be added to another vector, head to tail. The order of addition of two vectors does not matter, because the result will be the same. Check laws of vector addition for more details.

Vectors Definition

The vectors are defined as an object containing both magnitude and direction. Vector describes the movement of an object from one point to another.  Vector math can be geometrically picturised by the directed line segment.

The length of the segment of the directed line is called the magnitude of a vector and the angle at which the vector is inclined shows the direction of the vector. The starting point of a vector is called “Tail” and the ending point (having an arrow) is called “Head.”

A vector is defined as a mathematical structure. It has many applications in the field of physics and geometry. We know that the location of the points on the coordinate plane can be represented using the ordered pair such as (x, y). The usage of the vector is very useful in the simplification process of three-dimensional geometry.

Along with the term vector, we have heard the term scalar. A scalar actually represents the “real numbers”. In simpler words, a vector of “n” dimensions is an ordered collection of n elements called “ components “.

Examples of Vectors

The most common examples of the vector are Velocity, Acceleration, Force, Increase/Decrease in Temperature etc. All these quantities have directions and magnitude both. Therefore, it is necessary to calculate them in their vector form.

Also, speed is a quantity that has magnitude but no direction. This is the basic difference between speed and velocity.

Vector Notation

As we know already, a vector has both magnitude and direction. In the above figure, the length of the line AB is the magnitude and head of the arrow points towards the direction.

Magnitude of a Vector

The magnitude of a vector is shown by vertical lines on both the sides of the given vector “|a|”. It represents the length of the vector. Mathematically, the magnitude of a vector is calculated by the help of “ Pythagoras Theorem ,” i.e.

|a|= √(x 2 +y 2 )

Unit Vector

A unit vector has a length (or magnitude) equal to one, which is basically used to show the direction of any vector. A unit vector is equal to the ratio of a vector and its magnitude. Symbolically, it is represented by a cap or hat (^). 

If a is vector of arbitrary length and its magnitude is ||a||, then the unit vector is given by:

It is also known as normalising a vector.

Zero Vector

A vector with zero magnitudes is called a zero vector. The coordinates of zero vector are given by (0,0,0) and it is usually represented by 0 with an arrow (→) at the top or just 0. 

The sum of any vector with zero vector is equal to the vector itself, i.e., if ‘a’ is any vector, then;

Note: There is no unit vector for zero vector and it cannot be normalised.

Operations on Vectors

In maths, we have learned the different operations we perform on numbers. Let us learn here the vector operation such as Addition, Subtraction, Multiplication on vectors.

Addition of Vectors

The two vectors a and b can be added giving the sum to be a + b. This requires joining them head to tail.

We can translate the vector b till its tail meets the head of a. The line segment that is directed from the tail of vector a to the head of vector b is the vector “a + b”.

Characteristics of Vector Math Addition

• Commutative Law- the order of addition does not matter, i.e, a + b = b + a
• Associative law- the sum of three vectors has nothing to do with which pair of the vectors are added at the beginning.

i.e. (a + b) + c = a + (b + c)

Subtraction of Vectors

Before going to the operation it is necessary to know about the reverse vector(-a).

A reverse vector (-a) which is opposite of ‘a’ has a similar magnitude as ‘a’ but pointed in the opposite direction.

First, we find the reverse vector.

Then add them as the usual addition.

Such as if we want to find vector b – a

Then, b – a = b + (-a)

Scalar Multiplication of Vectors

Multiplication of a vector by a scalar quantity is called “Scaling.” In this type of multiplication, only the magnitude of a vector is changed not the direction.

• S(a+b) = Sa + Sb
• (S+T)a = Sa + Ta
• a.(-1) = -a

Scalar Triple Product

The scalar triple product , also called a box product or mixed triple product, of three vectors, say a, b and c is given by (a×b)⋅c . Since it involves dot product and evaluates single value, therefore stated as the scalar product. It is also denoted by ( a b c ). 

( a b c ) = (a×b)⋅c

\(\begin{array}{l}\begin{aligned} (\mathbf{a} \times \mathbf{b}) \cdot \mathbf{c} &=\left|\begin{array}{ll} a_{2} & a_{3} \\ b_{2} & b_{3} \end{array}\right| c_{1}-\left|\begin{array}{ll} a_{1} & a_{3} \\ b_{1} & b_{3} \end{array}\right| c_{2}+\left|\begin{array}{ll} a_{1} & a_{2} \\ b_{1} & b_{2} \end{array}\right| c_{3} \\ &=\left|\begin{array}{lll} c_{1} & c_{2} & c_{3} \\ a_{1} & a_{2} & a_{3} \\ b_{1} & b_{2} & b_{3} \end{array}\right| \end{aligned}\end{array} \)

The major application of the scalar triple product can be seen while determining the volume of a parallelepiped , which is equal to the absolute value of |(a×b)⋅c|, where a, b and c are the vectors denoting the sides of parallelepiped respectively. Hence,

Volume of parallelepiped = ∥a×b∥ ∥c∥ |cosϕ|=|(a×b)⋅c|

Vector Multiplication

Basically, there are two types of vector multiplication:

• Cross product
• Dot product

Cross Product of Vectors

The cross product of two vectors results in a vector quantity. It is represented by a cross sign between two vectors.

i.e., a × b

The mathematical value of a cross product-

| a |  is the magnitude of vector a.

| b | is the magnitude of vector b.

θ is the angle between two vectors a & b.

Dot product of  Vectors

The dot product of two vectors always results in scalar quantity, i.e. it has only magnitude and no direction. It is represented by a dot (.) in between two vectors.

a dot b = a. b

The mathematical value of the dot product is given as

Components of Vectors (Horizontal & Vertical)

There are two components of a vector in the x-y plane.

• Horizontal Component
• Vertical Component

Breaking a vector into its x and y components in the vector space is the most common way for solving vectors.

A vector “a” is inclined with horizontal having an angle equal to θ.

This given vector “a” can be broken down into two components i.e. a x and a y .

The component a x is called a “Horizontal component” whose value is a cos θ .

The component a y is called a “Vertical component” whose value is a sin θ.

Applications of Vectors

Some of the important applications of vectors in real life are listed below:

• The direction in which the force is applied to move the object can be found using vectors.
• To understand how gravity uses a force of attraction on an object to work.
• The motion of a body which is confined to a plane can be obtained using vectors.
• Vectors help in defining the force applied on a body simultaneously in the three dimensions. 
• Vectors are used in the field of Engineering, where the force is much stronger than the structure will sustain, else it will collapse.
• In various oscillators, vectors are used.
• Vectors also have its applications in ‘Quantum Mechanics’.
• The velocity in a pipe can be determined in terms of the vector field—for example, fluid mechanics.
• We may also observe them everywhere in the general relativity.
• Vectors are used in various wave propagations such as vibration propagation, sound propagation, AC wave propagation, and so on.

Problems and Solutions

Example 1: Given vector V, having a magnitude of 10 units & inclined at 60° . Break down the given vector into its two components.

Given, Vector V  having magnitude|V| = 10 units and θ  = 60°

Horizontal component (V x ) = V cos θ

V x = 10 cos 60°

V x = 10 × 0.5

V x = 5 units

Now, Vertical component(V y ) = V sin θ

V y = 10 sin 60°

V y = 10 × √3/2

V y = 10√3  units

Find the magnitude of vector a (3,4).

Given Vector a  = (3,4)

|a|= √(3 2 +4 2 )

|a|= √ (9+16)  = √25

Therefore, | a |= 5

Find the scalar and vector multiplication of two vectors ‘a’ and ‘b’, given by 3i – 1j + 2k and 1i + -2j + 3k respectively.

Given vector a (3,-1,2) and vector b (1,-2,3)

Where θ is the angle between the vectors. But we don’t know the angle between the vectors thus another method of multiplication can be used.

a.b = (3i – 1j + 2k) . (1i -2j +3k)

a.b = 3(i.i) + 2(j.j) + 6(k.k)

a.b = 3 + 2 + 6

Frequently Asked Questions on Vectors – FAQs

What is a vector in maths.

A vector is a quantity which has both magnitude and direction. It defines the movement of the object from one point to another.

What are the examples of vectors?

The examples of vectors are force, velocity, acceleration, etc., since these quantities have both magnitude and direction.

What are the types of vectors?

The ten types of vectors in Maths are: Zero Vector Unit Vector Position Vector Co-initial Vector Like and Unlike Vectors Co-planar Vector Collinear Vector Equal Vector Displacement Vector Negative of a Vector

What is the magnitude of the vector?

The magnitude of a vector is shown by vertical lines on both the sides of the given vector “|a|”. It represents the length of the vector.

What is the difference between scalar and vector?

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