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 ,….