# Thread: Cross Product proof?

1. ## Cross Product proof?

I have to prove that (a - b) x (a + b) = 2(a x b)

I get this:

(a + -b) x c = 2(a x b)

a x c + -b x c = 2(a x b)

a x (a + b) + -b x (a + b)= 2(a x b)

a x a + a x b + -b x a + -b x b= 2( a x b) - (a x b)
-(a x b)

0 + -b x a + -b x b = a x b

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

-b x b = 0

and I can't do anything

Thanks for your help

2. Originally Posted by Warrenx
I have to prove that (a - b) x (a + b) = 2(a x b)

I get this:

(a + -b) x c = 2(a x b)

a x c + -b x c = 2(a x b)

a x (a + b) + -b x (a + b)= 2(a x b)

a x a + a x b + -b x a + -b x b= 2( a x b) - (a x b)
-(a x b)

0 + -b x a + -b x b = a x b

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

-b x b = 0

and I can't do anything

Thanks for your help
$\mathbf{a} = x_1\mathbf{i} + y_1\mathbf{j} + z_1\mathbf{k}$

$\mathbf{b} = x_2\mathbf{i} + y_2\mathbf{j} + z_2\mathbf{k}$.

$\mathbf{a} + \mathbf{b} = (x_1 + x_2)\mathbf{i} + (y_1 + y_2)\mathbf{j} + (z_1 + z_2)\mathbf{k}$.

$\mathbf{a} - \mathbf{b} = (x_1 - x_2)\mathbf{i} + (y_1 - y_2)\mathbf{j} + (z_1 - z_2)\mathbf{k}$.

$(\mathbf{a} + \mathbf{b}) \times (\mathbf{a} - \mathbf{b}) = \left|\begin{matrix}\mathbf{i} & \mathbf{j} & \mathbf{k} \\ x_1 + x_2 & y_1 + y_2 & z_1 + z_2 \\ x_1 - x_2 & y_1 - y_2 & z_1 - z_2\end{matrix}\right|$

Go from here...

3. Originally Posted by Warrenx
I have to prove that (a - b) x (a + b) = 2(a x b)

I get this:

(a + -b) x c = 2(a x b)

a x c + -b x c = 2(a x b)

a x (a + b) + -b x (a + b)= 2(a x b)

a x a + a x b + -b x a + -b x b= 2( a x b) - (a x b)
-(a x b)
Are you saying that you are subtracting (a x b) from both sides?

There is no need to do that. Yes, you have, on the left
a x a + a x b - b x a+ b x b

But the cross product is anti-symmetric: a x a= b x b= 0 and
b x a= - a x b so
a x a+ a x b- b x a + b x b= 0+ a x b- (-a x b)+ 0= 2 a x b.

0 + -b x a + -b x b = a x b

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

-b x b = 0

and I can't do anything

Thanks for your help
You can't do anything because you went ahead and canceled everything and arrived at the trivially true statement "-b x b = 0"!

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