Chapter 11, Problem 37RE

To determine

Find the given vectors are orthogonal, parallel or neither

Answer to Problem 37RE

The vectors $\text{u =}\langle 39,-12,21\rangle$ and $\text{v }=\langle -26,8,-14\rangle$ are not orthogonal.

Explanation of Solution

Given:

The vectors

  $\text{u =}\langle 39,-12,21\rangle$ and $\text{v }=\langle -26,8,-14\rangle$

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

  $\text{u =}\langle 39,-12,21\rangle$ and

  $\text{v }=\langle -26,8,-14\rangle$

The two vectors u and v are parallel to each other as they have the relation u = cv where cis

some constant.

Therefore,

  $\begin{array}{l}\text{u =}\langle 39,-12,21\rangle \\ =-\frac{3}{2}\langle -26,8,-14\rangle \\ =-\frac{3}{2}\text{v}\end{array}$

And hence the vectors $\text{u =}\langle 39,-12,21\rangle$ and $\text{v }=\langle -26,8,-14\rangle$ are parallel.

Now to check whether the vectors $\text{u =}\langle 39,-12,21\rangle$ and $\text{v }=\langle -26,8,-14\rangle$

are orthogonal or not.

  $\begin{array}{l}\text{u}\text{.v }=\langle 39,-12,21\rangle ·\langle -26,8,-14\rangle \hfill \\ \begin{array}{l}\begin{array}{l}=\left(39\right)\left(-26\right)+\left(-12\right)\left(8\right)+\left(21\right)\left(-14\right)\hfill \\ =-1014-96-294\hfill \end{array}\\ =-1404\end{array}\hfill \\ \hfill \\ \hfill \end{array}$

Hence the vectors $\text{u =}\langle 39,-12,21\rangle$ and $\text{v }=\langle -26,8,-14\rangle$ are not orthogonal.