Chapter 6.3, Problem 21E

To determine

To calculate:

The following two vectors are equivalent or not:

  $\begin{array}{l}u:\text{}\left(x_1,y_1\right)=\left(2,-1\right),\left(x_2,y_2\right)=\left(5,-10\right)\\ v:\text{}\left(x_1,y_1\right)=\left(6,1\right),\left(x_2,y_2\right)=\left(9,-8\right)\end{array}$

Answer to Problem 21E

Vectors $u$ and $v$ are equivalent.

Explanation of Solution

Given information:

  $\begin{array}{l}u:\text{}\left(x_1,y_1\right)=\left(2,-1\right),\left(x_2,y_2\right)=\left(5,-10\right)\\ v:\text{}\left(x_1,y_1\right)=\left(6,1\right),\left(x_2,y_2\right)=\left(9,-8\right)\end{array}$

Formula used:

The magnitude of vector is given by the following expression:

  $d=\sqrt{{\left(x_2-x_1\right)}^2+{\left(y_2-y_1\right)}^2}$

The direction of vector or slope value is given by the following expression:

  $m=\frac{y_2-y_1}{x_2-x_1}$

Calculation:

The magnitude of vector $u$ is calculated as:

  $\begin{array}{l}\Vert u\Vert =\sqrt{{\left(2-5\right)}^2+{\left(-10+1\right)}^2}\\ \Vert u\Vert =\sqrt{9+81}\\ \Vert u\Vert =\sqrt{90}\end{array}$

The magnitude of vector $v$ is calculated as:

  $\begin{array}{l}\Vert v\Vert =\sqrt{{\left(6-9\right)}^2+{\left(1+8\right)}^2}\\ \Vert v\Vert =\sqrt{9+81}\\ \Vert v\Vert =\sqrt{90}\end{array}$

The slope value of vector $u$ is calculated as:

  $\begin{array}{l}{m}_u=\frac{y_2-y_1}{x_2-x_1}\\ m_u=\frac{-10+1}{5-2}\\ m_u=-\frac{3}{1}\end{array}$

The slope value of vector $v$ is calculated as:

  $\begin{array}{l}{m}_v=\frac{y_2-y_1}{x_2-x_1}\\ m_v=\frac{-8-1}{9-6}\\ m_v=-\frac{3}{1}\end{array}$

So, the values of the magnitudes and slope value of vector $u$ and $v$ are same therefore both vectors are equivalent.