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Vector Algebra

Vector and its Related Concepts


  • 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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