Chapter 11, Problem 38RE

To determine

Find the given vectors are orthogonal, parallel or neither

Answer to Problem 38RE

The vectors $\text{u =}\langle 8,5,-8\rangle$ and $\text{v }=\langle -2,4,-\frac{1}{2}\rangle$ are orthogonal.

Explanation of Solution

Given:

The vectors

  $\text{u =}\langle 8,5,-8\rangle$ and $\text{v }=\langle -2,4,-\frac{1}{2}\rangle$

The two vectors $\text{u }=\langle x_1,y_1,z_1\rangle$ and $\text{v}=\langle x_2,y_2,z_2\rangle$ are parallel to each other as they have therelation $\text{u}=c\text{v}$ where cis some constant.

The vectors $\text{u }=\langle x_1,y_1,z_1\rangle$ and $\text{v}=\langle x_2,y_2,z_2\rangle$ are orthogonal to each other if the dot product of $\text{u }=\langle x_1,y_1,z_1\rangle$ and $\text{v}=\langle x_2,y_2,z_2\rangle$ is 0 thatis $\text{u}\text{.v}=0$ .

Consider the two vectors $\text{u }=\langle x_1,y_1,z_1\rangle$ and $\text{v}=\langle x_2,y_2,z_2\rangle$ then the dot product of u and v is

  $\begin{array}{l}\text{u}\text{.v }=\langle x_1,y_1,z_1\rangle ·\langle x_2,y_2,z_2\rangle \hfill \\ =x_1\left(x_2\right)+y_1\left(y_2\right)+z_1\left(z_2\right)\hfill \\ \hfill \\ \hfill \end{array}$

Consider the following vectors

  $\text{u =}\langle 8,5,-8\rangle$ and

  $\text{v }=\langle -2,4,-\frac{1}{2}\rangle$

Now to check whether the vectors $\text{u =}\langle 8,5,-8\rangle$ and $\text{v }=\langle -2,4,-\frac{1}{2}\rangle$

The above two vectors u and v are not parallel to each other, because if $\text{u}=c\text{v}$ for some scalar c then the vectors are said to be parallel and there is no relation of the type $\text{u}=c\text{v}$ between u and v therefore, u and v are not parallel.

To check whether the vectors are orthogonal or not, first compute the dot product of u and v and if itis 0 then u and v will be orthogonal to each other but in this problem the dot product is not equal to 0 as

  $\begin{array}{l}\text{u}\text{.v }=\langle 8,5,-8\rangle ·\langle -2,4,-\frac{1}{2}\rangle \hfill \\ \begin{array}{l}=(8)(-2)+5(4)+(-8)\left(\frac{1}{2}\right)\\ =-16+20-4\\ =0\end{array}\hfill \\ \hfill \\ \hfill \end{array}$

Hence the vectors $\text{u =}\langle 8,5,-8\rangle$ and $\text{v }=\langle -2,4,-\frac{1}{2}\rangle$ are orthogonal.