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## Homework Statement

Prove that $$\bf{ a \times ( b \times c ) = \phi [ b(a \bullet c) - c(a \bullet b) ]} $$

for some constant phi

## Homework Equations

## The Attempt at a Solution

So I have used the unit vectors i, j, and k and found out that phi = 1.

With the main part of the proof, we are not allowed to expand it using components. Also, I am not at the level where we know the tensors or more advanced methods that I have seen elsewhere on the internet- just the basic rules of dot and cross product.

So I know that $$ a \times ( b \times c ) = p c + q b + k(b \times c)$$ for some scalars p,q, and k. I also know that the a x ( b x c) will lie in the plane of bc, so thus k = 0.

$$ \textbf{a} \times ( \textbf{b} \times \textbf{c} ) = p \textbf{c} + q \textbf{b} $$

Our professor then gave the hint of dotting both sides by a and (b x c) separately and going from there.

So when I dotted both sides by a, I got:

$$ 0 = p (\bf{a \bullet c}) + q (\bf{a \bullet b}) $$

Then when I dotted everything by (b x c):

$$ 0 = p (\bf{c \bullet (b \times c)}) + q (\bf{b \bullet (b \times c)}) $$

This is where I don't know what to do, because the second expression goes to 0 = 0 and I cannot really see how to use it.

Any help is greatly appreciated.