Simplify $A \times (A \times B)$

Question

Where $A$ and $B$ are vectors, and $\times$ is the cross product operator. I was able to get $A(A \cdot B) - B$ using the vector triple product, but it doesn't look like a simplified version to me.

Answer 1

Firstly note that, the vector triple product you want to evaluate is actually: $$A \times (A \times B)=(A\cdot B)A-(|A|^2)B$$

And this is, in fact, the most simplified expression that we can arrive at with what little you have given us!

Try to prove a more general following result:

$$A \times (B \times C)=(A \cdot C)B-(A \cdot B)C$$

Hint: Take, $A=a_1\hat i+a_2 \hat j+a_3 \hat k$ and so on for other vectors and do brute force computation.