# How do you find the sum of the unit vectors (2,2,7) and (5, -6, 2)?

Jul 26, 2016

Since the given $\left(2 , 2 , 7\right)$ and $\left(5 , - 6 , 2\right)$ are not unit vectors it is not clear what was intended by this question.

#### Explanation:

Possibility 1: Sum of the unit vectors with the same orientation as the given values
Since $\sqrt{{2}^{2} + {2}^{2} + {7}^{2}} = \sqrt{57}$
the unit vector corresponding to $\left(2 , 2 , 7\right)$ is
$\textcolor{w h i t e}{\text{XXX}} \left(\frac{2}{\sqrt{57}} , \frac{2}{\sqrt{57}} , \frac{7}{\sqrt{57}}\right)$

Similarly the unit vector corresponding to $\left(5 , - 6 , 2\right)$ is
$\textcolor{w h i t e}{\text{XXX}} \left(\frac{5}{\sqrt{65}} , - \frac{6}{\sqrt{65}} , \frac{2}{\sqrt{65}}\right)$

The sum of these unit vectors is
$\textcolor{w h i t e}{\text{XXX}} \left(\frac{2}{\sqrt{57}} + \frac{5}{\sqrt{65}} , \frac{2}{\sqrt{57}} - \frac{6}{\sqrt{65}} , \frac{7}{\sqrt{57}} + \frac{2}{\sqrt{165}}\right)$
(these terms could be evaluated but given the question's ambiguity I have not bothered with the effort involved)

Possibility 2: Unit vector with the same orientation as the sum of the given vectors
$\left(2 , 2 , 7\right) + \left(5 , - 6 , 2\right) = \left(7 , - 4 , 9\right)$
with corresponding unit vector:
$\textcolor{w h i t e}{\text{XXX}} \left(\frac{7}{\sqrt{146}} , - \frac{4}{\sqrt{146}} , \frac{9}{\sqrt{146}}\right)$