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

Vactor

Magnitude

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

MAGNITUDE

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 COSINES

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
DIRECTION RATIOS
DIRECTION RATIOS

Vectors

Equal Vectors

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

vector

Unit Vectors

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

Unit Vectors

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.

POSITION VECTOR OF A POINT

Negative Of A Vector

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

NEGATIVE OF A VECTOR

Components Of A Vector

COMPONENTS OF A VECTOR

Addition Of Vectors

Triangle Law

Triangle Law

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

Triangle Law

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,

Parallelogram Law

Multiplication Of A Vector By A Scalar

MULTIPLICATION OF A VECTOR BY A SCALAR

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

For Internal Division

For Internal Division

For External Division

For External Division

Properties And Application Of Scalar(Dot) Product Of Vectors

APPLICATION OF SCALAR

Properties Of Vector (Cross) Product Of Vectors

Properties Of Vector (Cross) Product Of Vectors