Chapter 6.3, Problem 27E

To determine

To calculate:

The component form and magnitude of following vector:

Initial point: $\left(-3,-5\right)$

Terminal point: $\left(-11,1\right)$

Answer to Problem 27E

The component form and magnitude of vector $v$ is $\langle -8,6\rangle$ and $10$ respectively.

Explanation of Solution

Given information:

Initial point: $\left(-3,-5\right)$

Terminal point: $\left(-11,1\right)$

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:

  $\begin{array}{l}\left(p_1,p_2\right)=\left(-3,-5\right)\\ \left(q_1,q_2\right)=\left(-11,1\right)\end{array}$

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

  $\begin{array}{l}{v}_1=q_1-p_1=-11+3=-8\\ v_2=q_2-p_2=1+5=6\end{array}$

So, the components of vector $v$ is $\langle v_1,v_2\rangle =\langle -8,6\rangle$

The magnitude of vector $v$ is calculated as:

  $\begin{array}{l}\Vert v\Vert =\sqrt{v_1{}^2+v_2{}^2}\\ \Vert v\Vert =\sqrt{{\left(-8\right)}^2+6^2}\\ \Vert v\Vert =\sqrt{64+36}\\ \Vert v\Vert =\sqrt{100}\\ \Vert v\Vert =10\end{array}$