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

Question

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

score 4 · Accepted Answer · answered Jan 17 '14 at 15:45

4

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).

answered Jan 17 '14 at 15:45

Brian Rushton

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An image to see what you mean? – user122081 Jan 17 '14 at 15:48
@user122081 See here:https://www.math.ucdavis.edu/~daddel/linear_algebra_appl/Applications/Determinant/Determinant/node4.html – Brian Rushton Jan 17 '14 at 15:49
https://www.math.ucdavis.edu/~daddel/linear_algebra_appl/Applications/Determinant/Determinant/img15.gif – Dan Rust Jan 17 '14 at 15:49
Haha, same link. Good ol' Google. – Dan Rust Jan 17 '14 at 15:50
I still don't understand :/ – user122081 Jan 17 '14 at 15:52
1

The area of that parallelogram, formed from the vectors $u$ and $v$ is the determinant $\det(u,v)$ (up to a change of sign which is dependant on the right hand rule). – Dan Rust Jan 17 '14 at 15:54
The idea is to put both vectors with there tails touching to get half of a parallelogram, and then draw in the other two edges. – Brian Rushton Jan 17 '14 at 15:56
Got it @DanielRust Thanks so much ! :D – user122081 Jan 17 '14 at 16:10

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

1 Answers1