Question

If $\mathbf{u}·\mathbf{v}=0$and $\mathit{u}\times \mathit{v}=0$, what, if anything, can you conclude about $u$ and $v$.

Short answer

We can conclude that at least one of the vectors$u,v=0$or have an angle of${90}^0,0^0$between them.

Step-by-step solution

Step 1. Dot product

We know,

$u·v=\left|u\right|\left|v\right|\cos \theta$

For $u·v=0$,

Either$$\begin{gathered} \left|u\right|=0 \\ \left|v\right|=0 \\ \cos \theta =0\Rightarrow \theta ={90}^0 \end{gathered}$$

Step 2. Cross Product

We know,

$\mathit{u}\times \mathit{v}=\left|u\right|\left|v\right|\sin \theta$

As given,

For $$\begin{gathered} \mathit{u}\times \mathit{v}=0 \\ \Rightarrow \left|u\right|=0or \\ \Rightarrow \left|v\right|=0or \\ \Rightarrow \sin \theta =0or\theta ={90}^0 \end{gathered}$$