Chapter 8.3, Problem 13E

To determine

To locate: the point $B$ in space and find magnitude of vector from origin to $B$ .

Answer to Problem 13E

  $\sqrt{69}$

Explanation of Solution

Given information:

  $B\left(7,2,4\right)$

Calculation:

The location of point in space can be done by using a geo-graphing calculator. The location of point $B$ in space is shown below.

Let the origin point be $O\left(0,0,0\right)$ and the line segment directed from $O$ to $B$ is vector $\overrightarrow{OB}$ . The ordered pair of $\overrightarrow{OB}$ is evaluated as follows:

  $\begin{array}{c}\overrightarrow{OB}=\langle x_2-x_1,y_2-y_1,z_2-z_1\rangle \\ =\langle 7-0,2-0,4-0\rangle \left[\text{Put ordered pairs of }O\text{ and }B\right]\\ =\langle 7,2,4\rangle \end{array}$

Thus the ordered pair is $\langle 7,2,4\rangle$ .

The magnitude of $\overrightarrow{OB}$ is evaluated as follows:

  $\begin{array}{c}\left|\overrightarrow{OB}\right|=\sqrt{{\left({\overrightarrow{OB}}_x\right)}^2+{\left({\overrightarrow{OB}}_y\right)}^2+{\left({\overrightarrow{OB}}_z\right)}^2}\\ =\sqrt{{\left(7\right)}^2+{\left(2\right)}^2+{\left(4\right)}^2}\left[\text{Put the ordered pairs of }\overrightarrow{OB}\right]\\ =\sqrt{49+4+16}\\ =\sqrt{69}\end{array}$

Thus the magnitude is $\sqrt{69}$ .