Solving Det(\vec{V} \otimes\vec{V})=0 in 10 Seconds: Tips and Tricks

[A-prime]

Homework Statement

The probem is to show within 10 seconds that Det(\vec{V} \otimes\vec{V})=0

with \vec{V}(x,y,z)=(V_x(\vec{r}),V_y(\vec{r}),V_z(\vec{r}))

The Attempt at a Solution

So I thought if I show the matrix is not invertible I would be done. But even with that I can't easily solve it in a few seconds...

Tips?

 

[fzero]

You can use the Levi-Civita symbol to write down the determinant. In 3d, this is

\det A = \sum_{ijk} \epsilon_{ijk} A_{1i} A_{2j} A_{3k}.

If you apply this to your dyadic, you should be able to see that it vanishes quickly.