# How do you find the magnitude of <-10,-5> and write it as a sum of the unit vectors?

Oct 23, 2017

$\left\mid \left\langle- 10 , - 5\right\rangle \right\mid = 5 \sqrt{5}$

$\left\langle- 10 , - 5\right\rangle = - 10 \boldsymbol{\underline{\hat{i}}} - 5 \boldsymbol{\underline{\hat{j}}}$

#### Explanation:

The magnitude is given by the metric norm:

$\left\mid \left\langle- 10 , - 5\right\rangle \right\mid = \sqrt{{\left(- 10\right)}^{2} + {\left(- 5\right)}^{2}}$
$\text{ } = \sqrt{100 + 25}$
$\text{ } = \sqrt{125}$
$\text{ } = 5 \sqrt{5}$

And using standard unit vector $\boldsymbol{\underline{\hat{i}}}$ and $\boldsymbol{\underline{\hat{j}}}$ in the $x$-direction and the $y$-direction respectively we have:

$\left\langle- 10 , - 5\right\rangle = - 10 \boldsymbol{\underline{\hat{i}}} - 5 \boldsymbol{\underline{\hat{j}}}$