Chapter 9, Problem 68RE

To determine

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

Answer to Problem 68RE

The value of the vectorsis: $\frac{-4}{5}i+\frac{3}{5}j$ .

Explanation of Solution

Given information:

The vectors:

  $v=-2i+j$ and $w=4i-3j$

Formula used:

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

To find the unit vector in the direction of opposite to the given vector multiply the unit vector by $-1$ .

Calculation:

Consider the vectors:

  $v=-2i+j$ and $w=4i-3j$

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

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

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

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

  $\begin{array}{c}\Vert w\Vert =\sqrt{{\left(4\right)}^2+{\left(-3\right)}^2}\\ =\sqrt{16+9}\\ =\sqrt{25}\\ =5\end{array}$

Apply the formula here to find the direction vector

  $\Vert w\Vert =\frac{w}{\Vert w\Vert }$

As from above: $\Vert w\Vert =5$

Then,

  $\begin{array}{l}\Vert w\Vert =\frac{4i-3j}{5}\\ \Vert w\Vert =\frac{4}{5}i-\frac{3}{5}j\end{array}$

Now to find the opposite direction of unit vector multiply the unit vector by $-1$ .

The unit vector in the opposite direction of the vector,

  $\begin{array}{l}\Vert w\Vert =\frac{4}{5}i-\frac{3}{5}j\\ \Vert w\Vert =-1\left(\frac{4}{5}i-\frac{3}{5}j\right)\\ \Vert w\Vert =-\frac{4}{5}i+\frac{3}{5}j\end{array}$ ,

Thus, the value of the magnitude of vectors are $-\frac{4}{5}i+\frac{3}{5}j$ .