Chapter 9, Problem 67RE

To determine

To calculate:The unit vector in the direction same as $v$ . Where the given vectors are $v=-2i+j$ and $w=4i-3j$ .

Answer to Problem 67RE

The value of the vectorsis: $\frac{-2}{\sqrt{5}}i+\frac{1}{\sqrt{5}}j$ .

Explanation of Solution

Given information:

The vectors:

  $v=-2i+j$ and $w=4i-3j$

Formula used:

The unit vector $\Vert v\Vert =\frac{v}{\Vert v\Vert }$ .

Calculation:

Consider the vectors:

  $v=-2i+j$ and $w=4i-3j$

Put the value of the vectors to find the unit vector $\Vert v\Vert =\frac{v}{\Vert v\Vert }$ .

Now $v=-2i+j$ and $w=4i-3j$ .

Now write the vectors $v=-2i+j$ and $w=4i-3j$ in $\Vert v\Vert =\frac{v}{\Vert v\Vert }$ .

To write the vectors $v=-2i+j$ and $w=4i-3j$ in this form first find $\Vert v\Vert$ .

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

Apply the formula here to find the direction vector

  $\Vert v\Vert =\frac{v}{\Vert v\Vert }$

As from above: $\Vert v\Vert =\sqrt{5}$

Then,

  $\begin{array}{l}\Vert v\Vert =\frac{-2i+j}{\sqrt{5}}\\ \Vert v\Vert =\frac{-2}{\sqrt{5}}i+\frac{1}{\sqrt{5}}j\end{array}$

Thus, the value of the magnitude of vectors are $\frac{-2}{\sqrt{5}}i+\frac{1}{\sqrt{5}}j$ .