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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

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The determinant is the area of the parallelogram spanned by the vectors, with a plus or minus depending on the order of the vectors.

If one is a multiple of the other, then that is like a parallelogram that is so skinny that its sides touch and it has no area.

It's useful in multivariable calculus for finding areas of things (e.g. change of variables or surface area).

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