Chapter 6.3, Problem 12E

To determine

To check whether $u$ and $v$ are equivalent.

Answer to Problem 12E

The vectors u and vare not equivalent.

Explanation of Solution

Given information:

The given vectors are u and v.

Initial point for uis $\left(2,0\right)$ .

Terminal point for uis $\left(7,4\right)$ .

Initial point for vis $\left(-8,1\right)$ .

Terminal point for vis $\left(2,9\right)$ .

Formula used:

The distance formula is,

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

Magnitude of a vector $u=xi+yj$ is

  $\Vert u\Vert =\sqrt{{\left(x\right)}^2+{\left(y\right)}^2}$

Calculation:

Consider the given points for vector u.

  $\left(2,0\right),\left(7,4\right)$

First find the magnitude of both the vectors from the distance formula.

Therefore, the magnitude of $u$ is,

  $\begin{array}{c}\Vert \text{u}\Vert =\sqrt{{\left(7-2\right)}^2+{\left(4-0\right)}^2}\\ =\sqrt{25+16}\\ =\sqrt{41}\end{array}$

Consider the given points for vector v.

  $\left(-8,1\right),\left(2,9\right)$

The magnitude of $v$ is,

  $\begin{array}{c}\Vert v\Vert =\sqrt{{\left(2+8\right)}^2+{\left(9-1\right)}^2}\\ =\sqrt{100+64}\\ =\sqrt{164}\end{array}$

Now,

  $\Vert u\Vert \ne \Vert v\Vert$

Magnitude of the vectors are not equal.

Therefore, the vectors are not equivalent.