Vectors

Vectors play an important role in mathematics, physics as well as in engineering. It is also known as Euclidean vector or geometric vector or spatial vector or simply “vectors”. According to vector algebra, a vector can be added to the other vector. Let us have a detailed discussion in vector math definition, representation, magnitude and its operations.

Vector 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 beginning point of a vector is called “Tail” and the end side (having arrow) is called “Head.”

Vectors

Vector Examples 

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.

Vector Math Representation

Vector Math Representation

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. Therefore, vectors between two points A and B is given as,vector ABor vector a. The arrow over the head of the vector shows the direction of the vector.

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|= √(x2+y2)

Vectors Operations

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.

Vector Addition

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

Vector Addition

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)

Vector Subtraction

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

Vector Subtraction

A reverse vector (-a) which is opposite of a has 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 wanna find vector b – a

Then, b – a = b + (-a)

Vector Multiplication

  • Scalar Multiplication

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.

  • Vector Multiplication

It is of two types “Cross product” and “Dot product.”

Cross Product

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

a × b

The mathematical value of a cross product-

Cross Product

where,

| a | is the magnitude of vector a.

| b | is the magnitude of vector b.

θ is the angle between two vectors a & b.

and \(\hat{n}\) is a unit vector showing the direction of the multiplication of two vectors.

Dot product

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

a. b

The mathematical value of the dot product is given as

a . b = | a | | b | cos θ

Vectors Components

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

Component of 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. ax and ay.

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

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

Vector Math 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 component.

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

Horizontal component (Vx) = V cos θ

Vx = 10 cos 60°

Vx = 10 × 0.5

Vx = 5 units

Now, Vertical component(Vy) = V sin θ

Vy = 10 sin 60°

Vy = 10 × √3/2

Vy = 10√3 units

 

Example 2- Find the magnitude of vector a (3,4).

Solution-

Given Vector a = (3,4)

|a|= √(x2+y2)

|a|= √(32+42)

|a|= √(9+16) = √25

Therefore, | a |= 5

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

Solution-

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

Vector Example

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

a.b = 11

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Practise This Question

Given the magnitude of vector A to be 6 units. Find the component of A along the x-axis?

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