Vector Algebra

Vector and its Related Concepts

Vector

The quantity that involves only magnitude (a value) is called a scalar quantity. Example: Length, mass, time, distance, etc.

The quantity that involves both magnitude and direction is called a vector. Example: Acceleration, momentum, force, etc.

Vector is represented as a directed line segment (line segment whose direction is given by means of an arrowhead).

In the following figure, line segment AB is directed towards B.

Hence, the vector representing directed line segment AB is or simply . Here, the arrow indicates the direction of AB. In , A is called the initial point and B is called the terminal point.

The position vector of a point P in space having coordinates (x, y, z) with respect to origin O (0, 0, 0) is given by or .

Here, the magnitude of i.e., || is given by.

If a position vector of point P (x, y, z) makes angles α, β, and γ with the positive directions of x−axis, y-axis and z-axis respectively, then these angles are called direction angles.

The cosine values of direction angles are called direction cosines of. This means that direction cosines (d.c.s.) of are cos α, cos β, and cos γ. We may write the d.c.s of as l, m, n where l = cos α, m = cos β and n = cos γ.

The direction ratios of will be lr, mr, and nr. We may write the direction ratios (d.r.s.) of as a, b, c, where a = lr, b = mr and c = nr.

If l, m, n are the d.c.s. of a position vector , then l2 + m2 + n2 = 1

 

Types of Vectors

A vector whose initial and terminal points coincide is called a zero vector or a null vector.

It is represented as .

A zero vector cannot be assigned in a definite direction since its magnitude is zero or it may be regarded as having any direction.

The vector , , etc. represents a zero vector.

A vector whose magnitude is unity or 1 unit is called a unit vector.

A unit vector in the direction of a position vector is given as .

Two or more vectors having the same initial point are called co-initial vectors.

In the following figure, vectors and are called initial vectors as each vector has the same initial point i.e., A.

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

In the following figure, and are collinear vectors.

Two vectors are said to be equal if they have the same magnitude and direction regardless of the positions of their initial points.

For two equal vectors and , we write =

A vector whose magnitude is the same as that of a given vector but whose direction is opposite to that of the given vector is called the negative of the given vector.

               The negative vector of is and it is written as = − .

​If a vector can be translated anywhere in the space without changing its magnitude and direction then such a vector is called free vector. For a vector of known magnitude and direction, if its initial point is fixed, then such a vector is known as localised vector. In other words, a vector which is drawn parallel to another vector through a specified point unlike free vector in space is called as Localised vector. Solved Examples

Example 1

Find the direction cosines and direction ratios of the position vector of point P(8, −4, 1).

Solution:

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