What is the meaning of $(\vec a \times \vec b)^2$?

Question

This is question given in my book:

Prove that– $$ (\vec a \times \vec b)^2 = \begin{vmatrix} \vec a \cdot \vec a & \vec a \cdot \vec b \\ \vec a \cdot \vec b & \vec b \cdot \vec b \end{vmatrix}$$

I want to know what is the meaning of $(\vec a \times \vec b)^2$ ? How can we square a vector ?

Answer 1

i think $$(\vec{a}\times \vec{b})^2=(\vec{a}\times \vec{b})\cdot (\vec{a}\times \vec{b})$$ where $\cdot $ means the dot-product

Answer 2

To square something you need to define a product. Because the result of the operation gives a scalar value, I think the author probably meant $\vec v^2=\vec v\cdot \vec v$, i.e. using the scalar product. I think it's a bit unfortunate, because the left hand side has a cross product. Not to mention that the rhs uses the notation $\vec a \cdot \vec a$ instead of $\vec a^2$, so at the very least, it is captious. Or as a comment from Ian suggests, it may be a typo: $|\vec a \times \vec b|^2$ makes sense.