How do you normalize (- 4i + 5 j- k)?

1 Answer
Apr 30, 2017

$- \frac{4}{\sqrt{42}} \hat{i} + \frac{5}{\sqrt{42}} \hat{j} - \frac{1}{\sqrt{42}} \hat{k}$

Explanation:

A normalised vector is just the vector divided by its metric norm, so that the norm of the new scaled vector is unity:

Let:

$\vec{u} = - 4 \hat{i} + 5 \hat{j} - \hat{k}$

So the metric norm is given by:

$| | \overline{u} | {|}^{2} = {\left(- 4\right)}^{2} + {\left(5\right)}^{2} + {\left(- 1\right)}^{2}$
$\text{ } = 16 + 25 + 1$
$\text{ } = 42$
$\therefore | | \overline{u} | | = \sqrt{42}$

And so:

$\hat{\overline{u}} = \frac{\overline{u}}{| | \overline{u} | |}$
$\setminus \setminus = \frac{\overline{u}}{\sqrt{42}}$
 \ \ = 1 / sqrt(42) (-4hati+5hatj-hatk
$\setminus \setminus = - \frac{4}{\sqrt{42}} \hat{i} + \frac{5}{\sqrt{42}} \hat{j} - \frac{1}{\sqrt{42}} \hat{k}$