How do you find all unit vectors orthogonal to both the vectors given below: (1,2,-1) and (3,3,-4)?

Jul 5, 2016

$\pm \frac{1}{\sqrt{35}} \left(- 5 , 1 , - 3\right)$

Explanation:

The unit vectors (in opposite directions) that are orthogonal to both

$a \mathmr{and} b$ is $\pm \frac{a X b}{|} a X b |$.

Here $a = \left(1 , 2 , - 1\right) \mathmr{and} b = \left(3 , 3 , - 4\right) ,$

$a X b = \left(\left(2\right) \left(- 4\right) - \left(3\right) \left(- 1\right) , \left(- 1\right) \left(3\right) - \left(- 4\right) \left(1\right) , \left(1\right) \left(3\right) - \left(3\right) \left(2\right)\right)$

$= \left(- 5 , 1 , - 3\right) \mathmr{and} | \left(- 5 , 1 , - 3\right) | = \sqrt{35}$.

So, the answer is $\pm \frac{1}{\sqrt{35}} \left(- 5 , 1 , - 3\right)$