# How do you find a unit vector that is orthogonal to both 2i+2j and 2i+2k where i,j, and k are vector use dot product?

Sep 18, 2016

See below.

#### Explanation:

Given $\vec{a} = \left(2 , 2 , 0\right)$ and $\vec{b} = \left(2 , 0 , 2\right)$ we can argue for a vector $\vec{v} = \left({v}_{1} , {v}_{2} , {v}_{3}\right)$ such that

$\left\lVert \vec{v} \right\rVert > 0$
$\left\langle\vec{a} , \vec{v}\right\rangle = 0$ and
$\left\langle\vec{b} , \vec{v}\right\rangle = 0$ resulting in the conditions

$\left\{\begin{matrix}{v}_{1}^{2} + {v}_{2}^{2} + {v}_{3}^{2} = {\left\lVert v \right\rVert}^{2} = 1 \\ 2 {v}_{1} + 2 {v}_{2} = 0 \\ 2 {v}_{1} + 2 {v}_{3} = 0\end{matrix}\right.$

solving this system regarding ${v}_{1} , {v}_{2} , {v}_{3}$ we obtain

$\vec{v} = \frac{1}{\sqrt{3}} \left(- 1 , 1 , 1\right)$