simplify the following experssions a b c are all vectors Sim

simplify the following experssions (a b c are all vectors)

Simplify the following experssions (a b c are all vectors) a times (b times (a times b)); (c) (a - b) middot (b - c) times (c - a);

Solution

Note : a x ( b x c ) = b( a.c ) - c( a.b )

(b) b x ( a x b ) = a( b.b ) - b( b.a )

Let , a.a = L , a.b = M and b.b = N

=> a x ( b x ( a x b ) ) = a x ( a( b.b ) - b( b.a ) )

=> a x ( b x ( a x b ) ) = ( a x Na ) - M( a x b )

As , a x a = 0

=> a x ( b x ( a x b ) ) = -M( a x b )

(c) (b - c) x (c - a) = (b x c) - (b x a) - (c x c) + (c x a)

=> (b - c) x (c - a) = (b x c) - (b x a) + (c x a)

=> (a - b) . (b - c) x (c - a) = a.( (b x c) - (b x a) + (c x a) ) - b.( (b x c) - (b x a) + (c x a) )

=> (a - b) . (b - c) x (c - a) = a.(b x c) - b.(c x a)

=> (a - b) . (b - c) x (c - a) = [ a b c ] - [ a b c ] = 0