Question

Find a unit vector in the same direction as $\mathit{v}=-2\mathit{i}+\mathit{j}$.

Short answer

The unit vector in the direction of$\mathit{v}=-2\mathit{i}+\mathit{j}$is $-\frac{2\sqrt{5}}{5}i+\frac{\sqrt{5}}{5}j$.

Step-by-step solution

Step 1. Given information

We have been given a vector $\mathit{v}=-2\mathit{i}+\mathit{j}$.

We have to find a unit vector in the same direction of v.

Step 2. Write the formula

Unit vector in the direction of $v=ai+bj$is given as:

$$\begin{gathered} u=\frac{v}{||v||} \\ ||v||=\sqrt{a^2+b^2} \end{gathered}$$

Step 3. Find the unit vector in the direction of v

$$\begin{gathered} \mathrm{u}=\frac{\mathrm{v}}{\parallel \mathrm{v}\parallel } \\ =\frac{-2\mathrm{i}+\mathrm{j}}{\parallel -2\mathrm{i}+\mathrm{j}\parallel } \\ =\frac{-2\mathrm{i}+\mathrm{j}}{\sqrt{(-2{)}^2+1^2}} \\ =\frac{-2\mathrm{i}+\mathrm{j}}{\sqrt{5}} \\ =\frac{-2}{\sqrt{5}}i+\frac{j}{\sqrt{5}} \\ =\frac{-2\sqrt{5}}{5}i+\frac{\sqrt{5}}{5}j \end{gathered}$$