Vector cross product properties?

Question

I have a 3x3 symmetric matrix

$$C=AB^T+BA^T,$$

where both $A$ and $B$ are 3x1 vectors.

How may I prove $$C(A\times B)=0?$$

I believe the key is the properties of vector cross product.

Answer 1

Hint. For any $3\times1$ vectors $u,v$ we have $$\def\v#1{{\bf#1}} \v u^T\v v=\v u\cdot\v v\ .$$ In particular, $$B^T(A\times B)=B\cdot(A\times B)\ .$$ There is a very important fact that you should know about this kind of expression involving both the dot and cross products.

Comment. My first equation is not correct, strictly speaking, since the LHS is a matrix and the RHS is a scalar. However, there is no important difference between a $1\times1$ matrix and a scalar.