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

Dec 29, 2016

Normalized vector is $- \frac{5}{\sqrt{66}} \hat{i} + \frac{4}{\sqrt{66}} \hat{j} - \frac{5}{\sqrt{66}} \hat{k}$

#### Explanation:

Normalizing a vector $\left(- 5 \hat{i} + 4 \hat{j} - 5 \hat{k}\right)$ means

a unit vector, whose initial point is same but terminal point is one unit in the same direction.

Hence, for a vector $\vec{v} = a \hat{i} + b \hat{j} + c \hat{k}$, its normalized vector is $\frac{1}{|} v | \left(a \hat{i} + b \hat{j} + c \hat{k}\right)$, where $| v | = \sqrt{{a}^{2} + {b}^{2} + {c}^{2}}$

and for $\left(- 5 \hat{i} + 4 \hat{j} - 5 \hat{k}\right)$, its normalized vector is

$\frac{1}{\sqrt{{\left(- 5\right)}^{2} + {4}^{2} + {\left(- 5\right)}^{2}}} \left(- 5 \hat{i} + 4 \hat{j} - 5 \hat{k}\right)$

= $\frac{1}{\sqrt{25 + 16 + 25}} \left(- 5 \hat{i} + 4 \hat{j} - 5 \hat{k}\right)$

= $- \frac{5}{\sqrt{66}} \hat{i} + \frac{4}{\sqrt{66}} \hat{j} - \frac{5}{\sqrt{66}} \hat{k}$