Chapter 11, Problem 10CLT

To determine

To Identify:

  $u$ and $v$ are orthogonal, parallel or neither.

Answer to Problem 10CLT

The vectors $u=\langle 4,-2,10\rangle$ and $v=\langle -2,6,2\rangle$ are orthogonal.

Explanation of Solution

Given Information:

  $\begin{array}{l}u=\langle 4,-2,10\rangle \\ v=\langle -2,6,2\rangle \end{array}$

Calculation:

Consider the vectors in space $a=\langle x_1,y_1,z_1\rangle$ and $b=\langle x_2,y_2,z_2\rangle$ then the dot product of

  $a=\langle x_1,y_1,z_1\rangle$ and $b=\langle x_2,y_2,z_2\rangle$ will be the vector

  $a.b=x_1{x}_2+y_1{y}_2+z_1{z}_2$

Two vectors $a=\langle x_1,y_1,z_1\rangle$ and $b=\langle x_2,y_2,z_2\rangle$ are said to be orthogonal if the dot product of $a=\langle x_1,y_1,z_1\rangle$ and $b=\langle x_2,y_2,z_2\rangle$ is $0$ that is if $a.b=0$ .

Consider the following vectors:

  $\begin{array}{l}u=\langle 4,-2,10\rangle \\ v=\langle -2,6,2\rangle \end{array}$

To determine whether the vectors are orthogonal, parallel or neither, proceed as follows.

Find the dot product of vectors. If it is zero, then the vectors are orthogonal.

Therefore,

  $\begin{array}{l}u.v=\left(\left(4\right)\left(-2\right)+\left(-2\right)\left(6\right)+\left(10\right)\left(2\right)\right)\\ =\left(-8-12+20\right)\\ =0\end{array}$

Since, the dot product of vector is equal to zero, vectors are orthogonal.