Vector Algebra
Vector and its Related Concepts
Vector
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The quantity that involves only magnitude (a value) is called a scalar quantity.
Example: Length, mass, time, distance, etc.
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The quantity that involves both magnitude and direction is called a vector.
Example: Acceleration, momentum, force, etc.
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Vector is represented as a directed line segment (line segment whose direction is given by means of an arrowhead).
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In the following figure, line segment AB is directed towards B.
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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.
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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
.
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Here, the magnitude of
i.e., |
| is given by
.
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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.
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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.
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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.
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If l, m, n are the d.c.s. of a position vector
, then
l2 + m2 + n2 = 1
Types of Vectors
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A vector whose initial and terminal points coincide is called a zero vector or a null vector.
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It is represented as
.
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A zero vector cannot be assigned in a definite d…
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