# The determinent of a vector? Where those it come from and then is is useful and true?

The determinant of a vector $\vec u$ and $\vec v$ is: $$\operatorname{det}(\vec{u},\vec{v})=\Big|\begin{matrix}a & c \\ b & d \end{matrix}\Big|=a\times d-b\times c$$

But what is it really? Where does it comes from and why is considered useful? Why is it true that if two vectors can satisfy this relation: $\vec v=k\times \vec u$ then there determinant is equal to 0?

Thank you