Question

If u and v are vectors in ${\mathrm{ℝ}}^3$ such that $u·v=0$ and $u\times v=0$, what can we conclude about u and v?

Short answer

$u·v=0\mathrm{and}u\times v=0$ concluded that at least one of them is$0$.

Step-by-step solution

Step 1. Given Information 

If u and v are vectors in ${\mathrm{ℝ}}^3$ such that $u·v=0$ and $u\times v=0$, what can we conclude about u andv?

Step 2. If u and v are vectors in ℝ3 such that u·v=0 and u×v=0.

$u·v=0$ if vectors u and v are orthogonal.

The cross product $u\times v=0$ is u and v are parallel.

Now we can also say that $u·v=0\mathrm{and}u\times v=0$concluded that at least one of them is $0$.