# How do you find two unit vectors orthogonal to A=(1, 3, 0) B =(2, 0, 5) first vector must have positive first coordinate?

Jul 1, 2016

$\pm \left(\frac{1}{\sqrt{286}}\right) \left(15 , - 5 , - 6\right)$

#### Explanation:

Let C(a, b, c) be a vector orthogonal to A(1, 3, 0) and B(2, 0, 5).

Then, $A . C = 0 = B . C$.

So, a+3b=0 and 2a+5b=0. Eliminating b and c,

C becomes $\left(a , - \frac{a}{3} , - 2 \frac{a}{5}\right) = a \left(1 , - \frac{1}{3} , - \frac{2}{5}\right)$

So, the unit vectors in opposite directions that are orthogonal to A

and B are

$\pm \left(\frac{1}{\sqrt{{1}^{2} + {\left(\frac{1}{3}\right)}^{2} + {\left(\frac{2}{5}\right)}^{2}}}\right) \left(1 , - \frac{1}{3} , - \frac{2}{5}\right)$

$= \pm \left(\frac{1}{\sqrt{286}}\right) \left(15 , - 5 , - 6\right)$