Vector Algebra

Quantities having magnitude only are called scalars. Such quantities which have magnitude and direction both are called vectors. A line segment which has a particular direction is called the directed line segment. A vector is a directed line segment. Vector OP is the position vector of P(x, y, z) with respect to origin O(0, 0, 0). cos , cos  and cos  are the direction cosines of vector r and are denoted by l, m and n respectively. A vector, whose initial and terminal points coincide, is called a zero (or null vector). A vector whose modulus is unity is called a unit vector. The unit vector in the direction of vector a is denoted by â. Thus, |â| = 1. Two or more vectors having the same initial point are called as co-initial vectors. Two or more vectors are said to be collinear if they are parallel to the same line, irrespective of their magnitudes and directions. Two vectors are said to be equal, if they have the same magnitude and direction regardless of the positions of their initial points. If two vectors are represented by two sides of the triangle taken in sequence, i.e., vectors are placed such that tail of a vector begins at the arrow head of vector placed at it, then the third side of the triangle in opposite direction of sequence, represents the sum or resultant of vector ‘a’ and vector ‘b’. If two vectors are represented in magnitude and direction by two adjacent sides of a parallelogram, then their sum is represented by the diagonal of the parallelogram which is co-initial with the given vectors. If vector a and vector b represents the adjacent sides of a triangle, then its area is equal to half of the modulus of the product of the two vectors. If vector a and vector b represents the adjacent sides of a parallelogram, then its area is equal to the modulus of the product of the two vectors. Keywords: Scalar Product, Section Formula, Direction Ratio, Component Vectors, Zero Vector, Vector Product, Position Vector, Zero Resultant.

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