Collinear Vectors - Simply Science

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Operations on Vectors

Two or more vectors are said to be parallel or collinear if they lie on a line or on parallel lines (irrespective of their magnitudes and directions).

Note: (i) Let AB = a, CD = b be two vectors. If A, B, C, D lie on a line or AB ∥ CD, then a, b are parallel. (ii) If a, b are parallel, then a, b have the same direction or opposite directions.

Relation between two collinear vectors:

If a and b are two parallel or collinear vectors, then there exists a scalar k such that a = kb If a = a1 i + a2 j + a3 k and b = b1 i + b2 j + b3 k are two parallel or collinear vectors, then

Test of collinearity of three points:

Three points whose position vectors are a, b, c are collinear iff there exists scalars x, y, z ( not all zero) such that x a + y b + z c = 0 where x + y + z = 0. If three points whose position vectors a = a1 i + a2 j, b = b1 i + b2 j and c = c1 i + c2 j are collinear, then If three points whose position vectors a = a1 i + a2 j + a3 k, b = b1 i + b2 j + b3 k and c = c1 i + c2 j + c3 k are collinear, then

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