Why is det(V ⊗ V) = 0 for any vector field V?

[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.