Chapter 9, Problem 35RE

To determine

To find: The unit vector in the same direction as $v$ .

Answer to Problem 35RE

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

Explanation of Solution

Given information:

The given vectors are

  $v=-2i+j$

  $w=4i-3j$

Calculation:

The magnitude of the vector $v=a_{1}i+b_{1}j$ .

  $\Vert v\Vert =\sqrt{a_1^2+b_1^2}$

Compare the vector $v=-2i+j$ with standard form $v=a_{1}i+b_{1}j$ to obtain the values of $a_1=-2$ and $b_1=1$ .

Calculate the magnitude of the vector.

  $\begin{array}{c}\Vert v\Vert =\sqrt{{\left(-2\right)}^2+{\left(1\right)}^1}\\ =\sqrt{4+1}\\ =\sqrt{5}\end{array}$

Calculate the unit vector in the same direction as $v$ .

  $\begin{array}{c}u=\frac{v}{\Vert v\Vert }\\ =\frac{-2i+j}{\sqrt{5}}\\ =\frac{-2i}{\sqrt{5}}+\frac{j}{\sqrt{5}}\end{array}$

Therefore, the unit vector in the same direction as $v$ is $\frac{-2i}{\sqrt{5}}+\frac{j}{\sqrt{5}}$ .