Question

For vectors u and v in${\mathrm{ℝ}}^3,\mathit{u}.(\mathit{u}\times \mathit{v})=\_\_\_\_\_\_\mathrm{and}\mathit{v}.(\mathit{u}\times \mathit{v})=\_\_\_\_\_\_$

Short answer

The required expressions are:

$\mathit{u}.(\mathit{u}\times \mathit{v})=0\mathrm{and}\mathit{v}.(\mathit{u}\times \mathit{v})=0$

Step-by-step solution

Given information

Two vectors v and u in${\mathrm{ℝ}}^3$.

Filling in the blanks

As cross product of two vectors is orthogonal to both the vectors:

$\mathit{u}.(\mathit{u}\times \mathit{v})=0\mathrm{and}\mathit{v}.(\mathit{u}\times \mathit{v})=0$