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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 d…

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