## Chapter:10

# What are Vectors ?

Vectors come in handy when we study those physical objects that have **direction as well as magnitude.** Let’s look at some of the terms associated with vectors :

## Scalar

It is a quantity which has magnitude but no direction. Examples : Mass, length, distance, speed etc.

## Vector

It is a quantity which has a magnitude as well as a specific direction. Examples : Weight, displacement, velocity etc. Denoted as

## Magnitude

The distance between initial and terminal points of a vector is called the magnitude of the vector.

## Direction Cosines

Direction cosines of a vector are the cosines of the angle between the vector and the three coordinate axes.

- As you see in the figure the angle α, β,γ made by the vector A with the positive directions of x,y and z-axis respectively, are called its direction angles. The cosine values of these angles ,i.e., cos α, cos β and cos γ are called direction cosines.
- Denoted by l,m and n.

## Direction Ratios

Numbers that are proportional to the direction cosines of the line are called direction ratios of the line.

- Ax = A cos α, Ay = A cos β and Az = A cos γ
- Denoted as a, b and c.
- Let (l, m, n) be the direction cosines of a line and direction ratios of the line be (a, b, c). Then

## Vectors

**Equal Vectors**

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

## Unit Vectors

If the magnitude of the vector is one , then it is called unit vector.

## Zero Vectors

If the magnitude of the vector is zero , then it is called null vector or zero vector. It can have any arbitrary direction.

## Parallel Vectors

Vectors which have same direction or exactly the opposite direction are called parallel vectors . The angle between them is 0° or 180°. Parallel vectors

are of two types :

## Like Parallel Vectors

If the angle between the vectors is 0°, then they are called like parallel vectors.

## Unlike Parallel Vectors

If the angle between the vectors is 180°, then they are called unlike parallel vectors.

## Collinear Vectors

Two or more vectors are said to be collinear vectors if they are parallel in same line irrespective of their magnitudes.

## Position Vector Of A Point

The vector OA is said to be a position vector of A with respect to Origin O, if A be any terminal point and O is the origin which is fixed.

## Negative Of A Vector

Two vectors are called negative vectors of each other if they have same magnitude but opposite direction. Denoted as

## Components Of A Vector

## Addition Of Vectors

**Triangle Law**

The addition of vectors is achieved by ‘tail to nose’ placing of the directed line segments in a triangle.

**Parallelogram Law**

The result of adding two co-initial vectors is the vector represented by the diagonal of the parallelogram formed with the component vectors as adjacent sides. eg: in the fig. below,

## Multiplication Of A Vector By A Scalar

## Position Vector Of A Point Dividing A Line Segment In A Given Ratio

**For Internal Division**

**For External Division**