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Math Insight

An introduction to vectors, definition of a vector.

A vector is an object that has both a magnitude and a direction. Geometrically, we can picture a vector as a directed line segment, whose length is the magnitude of the vector and with an arrow indicating the direction. The direction of the vector is from its tail to its head.

Two vectors are the same if they have the same magnitude and direction. This means that if we take a vector and translate it to a new position (without rotating it), then the vector we obtain at the end of this process is the same vector we had in the beginning.

Two examples of vectors are those that represent force and velocity. Both force and velocity are in a particular direction. The magnitude of the vector would indicate the strength of the force or the speed associated with the velocity.

We denote vectors using boldface as in $\vc{a}$ or $\vc{b}$. Especially when writing by hand where one cannot easily write in boldface, people will sometimes denote vectors using arrows as in $\vec{a}$ or $\vec{b}$, or they use other markings. We won't need to use arrows here. We denote the magnitude of the vector $\vc{a}$ by $\|\vc{a}\|$. When we want to refer to a number and stress that it is not a vector, we can call the number a scalar . We will denote scalars with italics, as in $a$ or $b$.

You can explore the concept of the magnitude and direction of a vector using the below applet. Note that moving the vector around doesn't change the vector, as the position of the vector doesn't affect the magnitude or the direction. But if you stretch or turn the vector by moving just its head or its tail, the magnitude or direction will change. (This applet also shows the coordinates of the vector, which you can read about in another page .)

The magnitude and direction of a vector. The blue arrow represents a vector $\vc{a}$. The two defining properties of a vector, magnitude and direction, are illustrated by a red bar and a green arrow, respectively. The length of the red bar is the magnitude $\|\vc{a}\|$ of the vector $\vc{a}$. The green arrow always has length one, but its direction is the direction of the vector $\vc{a}$. The one exception is when $\vc{a}$ is the zero vector (the only vector with zero magnitude), for which the direction is not defined. You can change either end of $\vc{a}$ by dragging it with your mouse. You can also move $\vc{a}$ by dragging the middle of the vector; however, changing the position of the $\vc{a}$ in this way does not change the vector, as its magnitude and direction remain unchanged.

More information about applet.

There is one important exception to vectors having a direction. The zero vector , denoted by a boldface $\vc{0}$, is the vector of zero length. Since it has no length, it is not pointing in any particular direction. There is only one vector of zero length, so we can speak of the zero vector.

Operations on vectors

We can define a number of operations on vectors geometrically without reference to any coordinate system. Here we define addition , subtraction , and multiplication by a scalar . On separate pages, we discuss two different ways to multiply two vectors together: the dot product and the cross product .

Addition of vectors

Given two vectors $\vc{a}$ and $\vc{b}$, we form their sum $\vc{a}+\vc{b}$, as follows. We translate the vector $\vc{b}$ until its tail coincides with the head of $\vc{a}$. (Recall such translation does not change a vector.) Then, the directed line segment from the tail of $\vc{a}$ to the head of $\vc{b}$ is the vector $\vc{a}+\vc{b}$.

The vector addition is the way forces and velocities combine. For example, if a car is travelling due north at 20 miles per hour and a child in the back seat behind the driver throws an object at 20 miles per hour toward his sibling who is sitting due east of him, then the velocity of the object (relative to the ground!) will be in a north-easterly direction. The velocity vectors form a right triangle, where the total velocity is the hypotenuse. Therefore, the total speed of the object (i.e., the magnitude of the velocity vector) is $\sqrt{20^2+20^2}=20\sqrt{2}$ miles per hour relative to the ground.

Addition of vectors satisfies two important properties.

The commutative law, which states the order of addition doesn't matter: $$\vc{a}+\vc{b}=\vc{b}+\vc{a}.$$ This law is also called the parallelogram law, as illustrated in the below image. Two of the edges of the parallelogram define $\vc{a}+\vc{b}$, and the other pair of edges define $\vc{b}+\vc{a}$. But, both sums are equal to the same diagonal of the parallelogram.

The associative law, which states that the sum of three vectors does not depend on which pair of vectors is added first: $$(\vc{a}+\vc{b})+\vc{c} = \vc{a} + (\vc{b}+\vc{c}).$$

You can explore the properties of vector addition with the following applet. (This applet also shows the coordinates of the vectors, which you can read about in another page .)

The sum of two vectors. The sum $\vc{a}+\vc{b}$ of the vector $\vc{a}$ (blue arrow) and the vector $\vc{b}$ (red arrow) is shown by the green arrow. As vectors are independent of their starting position, both blue arrows represent the same vector $\vc{a}$ and both red arrows represent the same vector $\vc{b}$. The sum $\vc{a}+\vc{b}$ can be formed by placing the tail of the vector $\vc{b}$ at the head of the vector $\vc{a}$. Equivalently, it can be formed by placing the tail of the vector $\vc{a}$ at the head of the vector $\vc{b}$. Both constructions together form a parallelogram, with the sum $\vc{a}+\vc{b}$ being a diagonal. (For this reason, the commutative law $\vc{a}+\vc{b}=\vc{b}+\vc{a}$ is sometimes called the parallelogram law.) You can change $\vc{a}$ and $\vc{b}$ by dragging the yellow points.

Vector subtraction

Before we define subtraction, we define the vector $-\vc{a}$, which is the opposite of $\vc{a}$. The vector $-\vc{a}$ is the vector with the same magnitude as $\vc{a}$ but that is pointed in the opposite direction.

We define subtraction as addition with the opposite of a vector: $$\vc{b}-\vc{a} = \vc{b} + (-\vc{a}).$$ This is equivalent to turning vector $\vc{a}$ around in the applying the above rules for addition. Can you see how the vector $\vc{x}$ in the below figure is equal to $\vc{b}-\vc{a}$? Notice how this is the same as stating that $\vc{a}+\vc{x}=\vc{b}$, just like with subtraction of scalar numbers.

Scalar multiplication

Given a vector $\vc{a}$ and a real number ( scalar ) $\lambda$, we can form the vector $\lambda\vc{a}$ as follows. If $\lambda$ is positive, then $\lambda\vc{a}$ is the vector whose direction is the same as the direction of $\vc{a}$ and whose length is $\lambda$ times the length of $\vc{a}$. In this case, multiplication by $\lambda$ simply stretches (if $\lambda>1$) or compresses (if $0 < \lambda <1$) the vector $\vc{a}$.

If, on the other hand, $\lambda$ is negative, then we have to take the opposite of $\vc{a}$ before stretching or compressing it. In other words, the vector $\lambda\vc{a}$ points in the opposite direction of $\vc{a}$, and the length of $\lambda\vc{a}$ is $|\lambda|$ times the length of $\vc{a}$. No matter the sign of $\lambda$, we observe that the magnitude of $\lambda\vc{a}$ is $|\lambda|$ times the magnitude of $\vc{a}$: $\| \lambda \vc{a}\| = |\lambda| \|\vc{a}\|$.

Scalar multiplications satisfies many of the same properties as the usual multiplication.

• $s(\vc{a}+\vc{b}) = s\vc{a} + s\vc{b}$ (distributive law, form 1)
• $(s+t)\vc{a} = s\vc{a}+t\vc{a}$ (distributive law, form 2)
• $1\vc{a} = \vc{a}$
• $(-1)\vc{a} = -\vc{a}$
• $0\vc{a} = \vc{0}$

In the last formula, the zero on the left is the number 0, while the zero on the right is the vector $\vc{0}$, which is the unique vector whose length is zero.

If $\vc{a} = \lambda\vc{b}$ for some scalar $\lambda$, then we say that the vectors $\vc{a}$ and $\vc{b}$ are parallel. If $\lambda$ is negative, some people say that $\vc{a}$ and $\vc{b}$ are anti-parallel, but we will not use that language.

We were able to describe vectors, vector addition, vector subtraction, and scalar multiplication without reference to any coordinate system. The advantage of such purely geometric reasoning is that our results hold generally, independent of any coordinate system in which the vectors live. However, sometimes it is useful to express vectors in terms of coordinates, as discussed in a page about vectors in the standard Cartesian coordinate systems in the plane and in three-dimensional space.

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Published 2004 Revised 2008

An Introduction to Vectors

Vectors are important in navigation where the actual velocity of an aeroplane relative to the earth is given by the combined velocities of the wind (which carries the plane along as if it were a glider) together with the velocity which the plane would have in still air. In the triangle above, if $\vec{OC}$ is along the direction required to reach the destination, and $\vec{AC}$ represents the velocity of the wind, then the pilot has to set a course in the direction $\vec{OA}$ with a speed calculated so that the sides of the triangle represent the velocities.

In mathematics we think of points and space as fundamental abstract concepts and we build a model of space by using a coordinate system. A three dimensional coordinate system is simply an infinite set of ordered triples of real numbers $(x, y, z)$ and each point is given by one of these ordered triples, called the coordinates of the point. To each free vector (or translation) there corresponds a position vector which is the image of the origin under that translation. So we define position vectors as points in space and to each position vector $P$ there corresponds a directed line segment $\vec {OP}$ which determines an infinite set of parallel directed line segments giving a unique free vector.

When we choose a coordinate system we effectively single out one representative from each free vector in space, namely the one which 'starts' from the chosen origin. If the point $A$ has coordinates $(x_a, y_a, z_a)$ then it has position vector ${\bf a} = (x_a, y_a, z_a)$ which may also be written as a column vector. There is a correspondence between the point $A$, the position vector ${\bf a}$ and the directed line segment $\vec {OA}$ which is a representative of the infinitely many segments making up a free vector.

It is frequently useful to work only with position vectors and not with free vectors. While there is a conceptual distinction between free vectors and position vectors it is possible to use both types interchangeably but this may cause confusion if we are not clear about the definitions.

All the vector algebra (adding, subtracting, multiplying) which works in one system corresponds to the vector algebra in the other system. When it suits us to do so we can switch from free vectors to position vectors or vice versa, do the vector algebra, then switch back with 'the answer'.

The magnitude of the position vector ${\bf a}=(x_a, y_a, z_a)$ is defined to be $$|{\bf a}|= (x_a^2+y_a^2+z_a^2)^{1/2}$$ and this is the length of the line segment $\vec{OA}$ and hence it is also the magnitude of the corresponding free vector.

The vectors ${\bf i}= (1, 0, 0)$, ${\bf j}=(0, 1, 0)$ and ${\bf k}=(0, 0, 1)$ are vectors of unit length parallel to the $x, y$ and $z$ axes. The position vector or point $A$ and the corresponding free vector consisting of all directed line segments parallel to $\vec {OA}$ can also be written as $x_a{\bf i}+y_a{\bf j}+z_a{\bf k}$.

Some elementary textbooks say that forces are vectors but are they? Strictly speaking they are a special type of vector with more structure than other vectors; as well as magnitude and direction forces are specified by their point or line of action. If I push your right shoulder hard enough you will turn one way and if I push your left shoulder with a force of the same magnitude in the same direction (an equal vector) you will turn the other way. The two forces have different turning effects so they are different forces even though they have the same 'vector properties'. When we add forces we simply use their vector properties but to specify a force we need to give its magnitude, direction and line of action.

Addition and subtraction of vectors

To add position vectors we simply add the components. For example if $\bf a$ is the position vector $(x_a, y_a, z_a)$ and $\bf b$ is the position vector $(x_b, y_b, z_b)$ then ${\bf a} + {\bf b} = (x_a+x_b,\ y_a+y_b,\ z_a+z_b).$ The parallelogram law is used to add position vectors giving $\vec{OA}+\vec{OB}=\vec{OC}$.

Note that, as a free vector ${\vec {OB}} = {\vec {AC}}$ so the parallelogram law of addition of position vectors exactly corresponds to the triangle law, ${\vec {OA}}+ {\vec {AC}} = {\vec {OC}}$, of addition of free vectors and hence they can be used interchangeably for either type of vector.

What about subtraction? Each point $A$ in space is a vector with components the same as the coordinates of the point, say ${\bf a}=(x_a,y_a,z_a)$. The reflection of the point $A$ in the origin is the point $A'$ with position vector $-{\bf a}=(-x_a,-y_a,-z_a)$. The effect of adding these two vectors is to give the zero vector. To subtract the vector ${\bf a}$ from the vector ${\bf c}$ simply add the vectors ${\bf c}$ and ${\bf -a}$.

The directed line segments $\vec{OA}$ and $\vec{OA'}$ are equal in length and opposite in direction so we say $\vec{OA'}=-\vec{OA}$. The equivalent method of subtraction for free vectors can be thought of as reversing the vector to be subtracted and adding it to the first vector. If $A$ and $C$ are two points $(x_a, y_a, z_a)$ and $(x_c, y_c, z_c)$ then the directed line segment ${\vec{OC}}= {\vec{OA}} + {\vec{AC}}$ so again we see that to subtract vectors we subtract the components. $${\vec {AC}}= {\vec {OC}} - {\vec {OA}} = \left( \begin{array}{cc} x_c-x_a\\ y_c-y_a\\ z_c-z_a \end{array} \right)$$

Multiplication of a vector by a scalar

We have already seen scalar multiples when we wrote $ (x_a, y_a, z_a) = x_a{\bf i}+y_a{\bf j}+z_a{\bf k}$. Here the vectors ${\bf i}, {\bf j}$ and ${\bf k}$ are multiplied by the scalars : $x_a, y_a$ and $z_a$.

One of the uses of multiplication of vectors by scalars is to write down an equation of a line using vectors. If a vector ${\bf d}$ is along a line then any other vector along the line is a multiple of ${\bf d}$ and we can call ${\bf d}$ a direction vector for the line. In writing down the equation of a line we use the notation ${\bf r} = (x,y,z)$ for the vector of a general point on the line, ${\bf e}=(x_1, y_1, z_1)$ for the vector of one particular point known to be on the line, $s$ as a scalar variable and ${\bf d} = (l,m,n)$ for a direction vector along the line. The vector equation of the line is then ${\bf r} = {\bf e} + s{\bf d}.$

-- Vectors - PureMaths 3 - RP AS & A Level Mathematics

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• Select activity Pure Mathematics 3 3.7 Vectors Pure Mathematics 3 3.7 Vectors

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Vector Mathematics

Jul 27, 2014

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Vector Mathematics. Physics 1. Physical Quantities. A scalar quantity is expressed in terms of magnitude (amount) only . Common examples include time, mass, volume, and temperature. Physical Quantities. A vector quantity is expressed in terms of both magnitude and direction.

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VectorMathematics Physics 1

Physical Quantities • A scalar quantity is expressed in terms of magnitude (amount) only. • Common examples include time, mass, volume, and temperature.

Physical Quantities • A vector quantity is expressed in terms of both magnitude and direction. • Common examples include velocity, weight (force), and acceleration.

Representing Vectors • Vector quantities can be graphically represented using arrows. • magnitude = length of the arrow • direction = arrowhead

Vectors • All vectors have a head and a tail.

Vector Addition • Vector quantities are addedgraphically by placing them head-to-tail.

Head-to-Tail Method • Draw the first COMPONENT vector with the proper length and orientation. • Draw the second COMPONENT vector with the proper length and orientation starting from the head of the first component vector.

Head-to-Tail Method • The RESULTANT (sum) vector is drawn starting at the tail of the first component vector and terminating at the head of the second component vector. • Measure the length and orientation of the resultant vector.

To add vectors, move tail to head and then draw resultant from original start to final point. East Resultant Resultant is (sqrt(2)) 45◦ south of East South

Vector addition is ‘commutative’ (can add vectors in either order) East Resultant Resultant is (sqrt(2)) 45◦ south of East South

Vector addition is ‘commutative’ (can add vectors in either order) East South Resultant Resultant is (sqrt(2)) 45◦ south of East Resultant South East

Co-linear vectors make a longer (or shorter) vector Resultant is 3 magnitude South

Can add multiple vectors. Just draw ‘head to tail’ for each vector Resultant is magnitude 45◦ North of East North East East North

Adding vectors is commutative. East North East North North Resultant is magnitude 45◦ North of East East East South

Equal but opposite vectors cancel each other out North West East Resultant=0 Resultant is 0. South

Vector Addition – same direction A + B = R A B B A R = A + B

Vector Addition • Example: What is the resultant vector of an object if it moved 5 m east, 5 m south, 5 m west and 5 m north?

Vector Addition – Opposite direction(Vector Subtraction) . A + (-B) = R A B -B A A + (-B) = R -B

Vectors • The sum of two or more vectors is called the resultant.

Practice Vector Simulator http://phet.colorado.edu/sims/vector-addition/vector-addition_en.html

Polar Vectors • Every vector has a magnitude and direction

Right Triangles SOH CAH TOA

Vector Resolution • Every vector quantity can be resolved into perpendicular components. • Rectilinear (component) form of vector:

Vector Resolution • Vector A has been resolved into two perpendicular components, Ax (horizontal component) and Ay (vertical component).

Vector Resolution • If these two components were added together, the resultant would be equal to vector A.

Vector Resolution • When resolving a vector graphically, first construct the horizontal component (Ax). Then construct the vertical component (Ay). • Using right triangle trigonometry, expressions for determining the magnitude of each component can be derived.

Vector Resolution • Horizontal Component (Ax)

Vector Resolution • Vertical Component (Ay)

Drawing Directions • EX: 30° S of W • Start at west axis and move south 30 ° • Degree is the angle between south and west N W E S

N (-,+) (+,+) E W (-,-) (+,-) S Vector Resolution • Use the sign conventions for the x-y coordinate system to determine the direction of each component.

Component Method • Resolve all vectors into horizontal and vertical components. • Find the sum of all horizontal components. Express as SX. • Find the sum of all vertical components. Express as SY.

Component Method • Construct a vector diagram using the component sums. The resultant of this sum is vector A + B. • Find the magnitude of the resultant vector A + B using the Pythagorean Theorem. • Find the direction of the resultant vector A + B using the tangent of an angle q.

Adding “Oblique” Vectors • Head to tail method works, but makes it very difficult to ‘understand’ the resultant vector

Adding “Oblique” Vectors • Break each vector into horizontal and vertical components. • Add co-linear vectors • Add resultant horizontal and vertical components

Adding “Oblique” Vectors • Break each vector into horizontal and vertical components.

Adding “Oblique” Vectors • Break each vector into horizontal and vertical components. • Add co-linear vectors

Using Calculator For Vectors • Can use the “Angle” button on TI-84 calculator to do vector mathematics

Using Calculator for Vectors

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• Preferences

Vector Mathematics - PowerPoint PPT Presentation

Vector Mathematics

... and the resultant is the straight line from start to finish ... check out the next s... components of vectors. a = ax ay. ax =a cos . ay = a sin ... – powerpoint ppt presentation.

• Adding, Subtracting, Multiplying and Dividing
• One can add 23 kg and 42 kg and get 65 kg.
• However, one cannot add together 23 m/s south and 42 m/s southeast and get 65 m/s south-southeast.
• Vectors addition takes into account adding both magnitude and direction
• Vector A measured quantity with both magnitude (the how big part) and direction
• Scalar A measured quantity with magnitude only
• Resultant Vector The final vector of a vector math problem
• Graphically Draw vectors to scale, Tip to Tail, and the resultant is the straight line from start to finish
• Mathematically Employ vector math analysis to solve for the resultant vector
• A 5.0 m _at_ 0
• B 5.0 m _at_ 90
• You can add vectors in any order and yield the same resultant.
• The math you used previously doesnt work (and I wont let you use the Law of Sines or Cosines) or does it???
• What we will do is break each vector into components
• The components are the x and y values of the polar coordinate (go back 6 slides)
• Check out the next slides
• As long as you draw the x component first
• We will organize these components in a table.
• See the board for this part and next slide
• Add all X components together ? Final Rx
• Add all Y components together ? Final Ry
• Simply add or subtract 180 (keep ? between 0 and 360) to the direction of the vector being subtracted
• You just ADD the OPPOSITE vector (there is no subtraction in vector math)
• A unit vector is a vector that has a magnitude of 1, with no units.
• Its only purpose is to point
• We will use i, j, k for our unit vectors
• i means x direction, j is y, and k is z
• We also put little hats () on i, j, k to show that they are unit vectors (I will boldface them)
• R (Ax Bx )i (Ay By )j (Az Bz )kwhich becomes R Rx i Ry j Rz k
• The magnitude of R is found by applying the Pythagorean theorem
• Scalar x Vector Vector w/ magnitude multiplied by the value of scalar A 5 m _at_ 303A 15m _at_ 30
• 2. (vector) (vector) Scalar
• This is called the Scalar Product or the Dot Product
• (vector) x (vector) vector
• This is called the vector product or the cross product
• You can also solve the Cross Product with a matrix and unit vectorscheck out the board for this.

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• 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?

Put your understanding of this concept to test by answering a few MCQs. Click ‘Start Quiz’ to begin!

Select the correct answer and click on the “Finish” button Check your score and answers at the end of the quiz

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apt4Maths: PowerPoint (Lesson 4 of 14) on Shapes, Angles & Construction - ANGLES - POLYGONS

Subject: Mathematics

Age range: 14-16

Resource type: Lesson (complete)

Last updated

6 August 2024

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PowerPoint Presentation on Angles – Polygons (20 slides): Reviews the names of common polygons, explains how to calculate exterior and interior angles of regular polygons as well as missing angles of irregular polygons, and explains what is meant by the term ‘tessellation’.

This is 1 of a set of 14 PowerPoints on ‘Shapes, Angles & Construction’. It has been written by a highly experienced teacher (of 25+ years), senior examiner and reviser for Maths and Stats examinations. It includes:

Lesson objectives Step-by-step explanations of the subject matter Examples to aid understanding Questions to check understanding Answers to questions, with explanations Suggestions regarding which topic(s) should be moved on to next.

The complete set of 14 PowerPoints (238 slides, excluding Title Pages) covers the following topics relating to ‘Shapes, Angles & Construction’:

01 Shapes – Terminology and Properties (18 slides). 02 Angles – An Introduction (25 slides). 03 Angles – Simple Facts (20 slides). 04 Angles – Polygons (20 slides). 05 Angles – Parallels (23 slides). 06 Angles – Bearings (9 slides). 07 Circle Theorems (21 slides). 08 Basic Trigonometry (24 slides). 09 Sine Rule (14 slides). 10 Scale Drawings (11 slides). 11 Constructions (13 slides). 12 Bisectors (11 slides). 13 Locus (14 slides). 14 Views (15 slides).

These PowerPoint Presentations are one of several sets of PowerPoint Presentations, which essentially relate to the ‘Geometry and Measures’ section of the Maths specifications. These other sets concern:

Measures, Perimeter, Area & Volume Symmetry, Transformations & Vectors

Note: Work on Pythagoras’ Theorem, the Cosine Rule, and the trigonometric formula for the area of a triangle, is included in APT’s set of PowerPoints on ‘Measures, Perimeter, Area & Volume’.

The purchase of this resource comes with a licence to make the resource available in digital and / or in print form (including photocopying) to the staff and students attending the purchasing institution, ie the individual school / college on a single site. The resource may be distributed via a secure virtual learning environment. It must not be made available on any public or insecure website or other platform. The resource must not be distributed to other institutions that are members of the same academy chain or similar organisation; each individual institution must purchase their own copy of the resource.

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