Chapter 6.3, Problem 11E

To determine

To show: u=v

Explanation of Solution

Given: Show that u is equivalent to v.

  

FORMULA USED:

Magnitude of vector: $\Vert v\Vert =\sqrt{{(x_2-x_1)}^2+{(y_2-y_1)}^2}$

Here for the vector u

  $\begin{array}{l}{x}_1=2,x_2=6\\ y_1=4,y_2=5\end{array}$

Then

The magnitude of vector u is given by

  $\Vert u\Vert =\sqrt{{(6-2)}^2+{(5-4)}^2}=\sqrt{17}$

For the vector v

  $\begin{array}{l}{x}_1=0,x_2=4\\ y_1=0,y_2=1\end{array}$

Then the magnitude of vector v is given by

  $\Vert v\Vert =\sqrt{{(4-0)}^2+{(1-0)}^2}=\sqrt{17}$

The magnitude of both the vectors is same

Now, finding the slope of each vector

  $\begin{array}{l}slop{e}_u=\frac{5-4}{6-2}=\frac{1}{4}\\ slop{e}_v=\frac{1-0}{4-0}=\frac{1}{4}\end{array}$

Since the slope are equal and they are pointing to the Right, and the direction is same

Since the direction and magnitude of both the vectors is same,They are equivalent.

Hence proved.