Reciprocal System of Vectors Let a, b and c be three non-coplanar vectors and let a’ = b * c / [ a b c ], b’ = c * a / [ a b c ], c’ = a * b / [ a b c ] Then, a’, b’ and c’ are said to form a reciprocal system of a, b and c.
Properties of Reciprocal System( i) a * a’ = b * b’= c * c’ = 1( ii) a * b’= a * c’ = 0, b * a’ = b * c’ = 0, c * a’ = c * b’= 0( iii) [ a’, b’, c’] [ a b c ] = 1 ⇒ [ a’ b’ c’] = 1 / [ a b c ]
( iv) a = b’ * c’ / [ a’, b’, c’], b = c’ * a’ / [ a’, b’, c’], c = a’ * b’ / [ a’, b’, c’] Thus, a, b, c is reciprocal to the system a’, b’, c’.
( v) The orthonormal vector triad i, j, k form self reciprocal system.
( vi) If a, b, c be a system of non-coplanar vectors and a’, b’, c’ be the reciprocal system of vectors, then any vector r can be expressed as r =( r * a’) a +( r * b’) b +( r * c’) c.
Linear Combination of Vectors
Let a, b, c,… be vectors and x, y, z, … be scalars, then the expression x a yb + z c + … is called a linear combination of vectors a, b, c,….
Collinearity of Three Points The necessary and sufficient condition that three points with PV’ s b, c are collinear is that there exist three scalars x, y, z not all zero such that xa + yb + zc ⇒ x + y + z = 0.
Coplanarity of Four Points
The necessary and sufficient condition that four points with PV’ s a, b, c, d are coplanar, if there exist scalar x, y, z, t not all zero, such that xa + yb + zc + td = 0 rArr; x + y + z + t = 0.
If r = xa + yb + zc … Then, the vector r is said to be a linear combination of vectors a, b, c,….