Question

If `vec A xx vec B= vec C xx vec B`, show that `vec C` need not be equal to vec A`.

Answer 1

` vec A xx vec B= vec C xx vec B or vec A xxx vec B -vec C xxx vec B=0`
or ` (vec A-vec C) xxx vec B=0` …(i)
To satisfy (i), the three possibilities can be there
(i) `vec A -vec C` =0 or `vec A =vec C`
(ii) ` vec B= vec 0`
(iii) ` vec A-vec C and vec B` are parallel to each other i.e. `vec A--vec C=n vec B`, wher (n) a non zeroreal number.
or `vec A = vec C + n vec B`
Thus, if ` vec A xx vec B =vec C xx vec B, vec C` need not be equal to `vec A`. The given statement is true if ` vec B`
is a zero vector of `vec A` is equal to vec C + n vec B`.