# What is the unit vector that is orthogonal to the plane containing  (29i-35j-17k)  and  (20j +31k) ?

Feb 9, 2016

The cross product is perpendicular to each of its factor vectors, and to the plane that contains the two vectors. Divide it by its own length to get a unit vector.

#### Explanation:

Find the cross product of

$v = 29 i - 35 j - 17 k$ ... and ... $w = 20 j + 31 k$

$v \times w = \left(29 , - 35 , - 17\right) \times \left(0 , 20 , 31\right)$

Compute this by doing the determinant $| \left(\begin{matrix}i & j & k \\ 29 & - 35 & - 17 \\ 0 & 20 & 31\end{matrix}\right) | .$

After you find $v \times w = \left(a , b , c\right) = a i + b j + c k ,$

then your unit normal vector can be either $n$ or $- n$ where

$n = \frac{v \times w}{\sqrt{{a}^{2} + {b}^{2} + {c}^{2}}} .$

You can do the arithmetic, right?

// dansmath is on your side! \\