# Simplifying Cross Products

• January 29th 2010, 07:18 AM
gotpho
Simplifying Cross Products
Hi guys

I'm boggled by this problem

a x (b x (a x b))

I know the first step is to simplify the triple vector product inside the () which leads this expression

a x (( b dot b)a - (b dot a)b)

but the book claims that the final answer is eventually

(- a dot ba x b)

Can someone explain why?
• January 29th 2010, 07:28 AM
Jester
Quote:

Originally Posted by gotpho
Hi guys

I'm boggled by this problem

a x (b x (a x b))

I know the first step is to simplify the triple vector product inside the () which leads this expression

a x (( b dot b)a - (b dot a)b)

but the book claims that the final answer is eventually

(- a dot ba x b)

Can someone explain why?

Expand your first step

$
b \cdot b \left( a \times a\right) - a \cdot b \left( a \times b\right)
$

noting that $a \times a = 0$.
• January 29th 2010, 09:03 AM
gotpho
Never knew you could distribute like that(Surprised)

But isn't a dot a = |a|^2. How do you know that they are orthogonal?
• January 29th 2010, 09:18 AM
Jester
Quote:

Originally Posted by gotpho
Never knew you could distribute like that(Surprised)

But isn't a dot a = |a|^2. How do you know that they are orthogonal?

I believe it's $a \times a$ not $a \cdot a.$
• January 29th 2010, 09:23 AM
gotpho
Quote:

Originally Posted by Danny
I believe it's $a \times a$ not $a \cdot a.$

Oh, my bad. Thanks.