Chapter 6.3, Problem 28E

To determine

To calculate:

The component form and magnitude of following vector:

Initial point: $\left(-2,7\right)$

Terminal point: $\left(5,-17\right)$

Answer to Problem 28E

The component form and magnitude of vector $v$ is $\langle 7,-24\rangle$ and $25$ respectively.

Explanation of Solution

Given information:

Initial point: $\left(-2,7\right)$

Terminal point: $\left(5,-17\right)$

Formula used:

The components of vector $v=\langle v_1,v_2\rangle$ are given by the following expression:

  $\begin{array}{l}{v}_1=q_1-p_1\\ v_2=q_2-p_2\end{array}$

The magnitude of vector $v$ is given by the following expression:

  $\Vert v\Vert =\sqrt{v_1{}^2+v_2{}^2}$

Calculation:

Know that:

  $\begin{array}{l}\left(p_1,p_2\right)=\left(-2,7\right)\\ \left(q_1,q_2\right)=\left(5,-17\right)\end{array}$

The components of vector $v=\langle v_1,v_2\rangle$ are calculated as:

  $\begin{array}{l}{v}_1=q_1-p_1=5+2=7\\ v_2=q_2-p_2=-17-7=-24\end{array}$

So, the components of vector $v$ is $\langle v_1,v_2\rangle =\langle 7,-24\rangle$

The magnitude of vector $v$ is calculated as:

  $\begin{array}{l}\Vert v\Vert =\sqrt{v_1{}^2+v_2{}^2}\\ \Vert v\Vert =\sqrt{7^2+{\left(-24\right)}^2}\\ \Vert v\Vert =\sqrt{49+576}\\ \Vert v\Vert =\sqrt{625}\\ \Vert v\Vert =25\end{array}$