Question

Prove that (a-b) x (a+b)=2a x b is true.
(a and b are vectors)

Answer 1

( a - b) x ( a + b ) = 2 a + b ( missing arrows above the letters- shows that a and b are vectors)
Left side:
(a x a) + (a x b) - (b x a) + (b x b)= ( a x b ) - ( b x a ) = ( a x b ) +( a x b )=
2 a x b
It is proved to be true.
Explanation: Vector product of the same vectors is 0 (a x a and b x b ). Also, vector product is anticommutative: a x b = - ( b x a ).

Answer 2

(a-b) X (a+b)
= aXa - bXa +aXb -bXb (distributing)

Now, cross product of a vector with itself = 0
so, aXa = 0, bXb = 0

Also, aXb = - bXa

so,
(a-b) X (a+b) = 0 + aXb + aXb + 0
= 2aXb

hence, proved :)