# How do you find a unit vector orthogonal to both vectors (1, -3, 2) and (-1, 2, 3)?

Feb 1, 2017

$\text{The Reqd. Unit Vector=} \frac{1}{\sqrt{195}} \left(- 13 , - 5 , - 1\right) .$

#### Explanation:

We know from Vector Geometry that $\vec{x} \times \vec{y}$ is a vector

which is orthogonal to both $\vec{x} \mathmr{and} \vec{y} .$

The desired Unit Vector , then, can be obtained as

$\frac{\vec{x} \times \vec{y}}{|} | \vec{x} \times \vec{y} | |$

$\text{Now, } \left(1 , - 3 , 2\right) \times \left(- 1 , 2 , 3\right) = | \left(i , j , k\right) , \left(1 , - 3 , 2\right) , \left(- 1 , 2 , 3\right) |$

$= \left(- 13 , - 5 , - 1\right) , \text{ so that, } | | \left(\left(- 13 , - 5 , - 1\right)\right) | | = \sqrt{195}$

$\text{Therefore, the Reqd. Unit Vector=} \frac{1}{\sqrt{195}} \left(- 13 , - 5 , - 1\right) .$