Question

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

Answer 1

here's what i got so far..
http://sketchtoy.com/56990936

Answer 2

@ybarrap can you please help me again? :)

Answer 3

A lot of use of cross product properties here...
You were right upto here:
((a-b) X a) + ((a-b) X b)
but since cross product is not commutative the distribution cannot be done in this state
however cross product is anti-commutative, where (m X n) = -(n X m)
so rewrite as -(a X (a-b)) + -(b X (a-b)); but we still can't distribute.
Cross product only exhibits distributivity on addition so rewrite again:
-(a X (a+(-b))) + -(b X (a+(-b)); now we can distribute
-( aXa + aX-b ) + -(b X a + bX-b); anything cross product with itself or its negation is a zero vector so get rid of those;
-(a X -b) + -( b X a); then using anti-commutativity on the second term:
-(a X -b) + (a X b);
now we must use the properties of scalar multiplication on cross products, where
r(a X b) = ra X b = a X rb, to distribute the negative through the first term to make the b positive
a X b + a X b = 2(aXb);

All done! There's probably better ways to do this but this is the solution I came up with

Answer 4

whoa, thanks a lot but i'm still confused with your solution because of the distributive property.. O.o
why is it that this property cannot be applied on the first part and you can do it on the last part?
sorry, i'm slow learner :/

Answer 5

$$
(a-b) \times (a+b)\\
=(a-b)\times a+(a-b)\times b\\
=a\times a -b\times a +a\times b-b\times b\\
=0+a\times b+a\times b-0\\
=2\left(a\times b\right )
$$
We used the distributive property of the cross product and the negative of a cross product is the positive cross product with the arguments reversed. Also, the cross product of parallel vectors is zero.

Let me know if you have any questions.

Answer 6

aahh.. got it! thanks @ybarrap and @mascoJ92