Chapter 10.2, Problem 54E

To determine

To check: The given statement is true or false.

Answer to Problem 54E

The given statement is false.

Explanation of Solution

Given information:

The statement is:

The vector with direction angle $0^0$ is added to the vector with direction angle ${90}^0$ .

The result of the vector with direction angle ${45}^0$ .

Calculation:

Know that the direction angle $\theta$ of a vector $r=v_1,v_2$ with the positive $x$ -axis,

  $\begin{array}{c}\sin \theta =\frac{v_2}{\left|v\right|}\\ \cos \theta =\frac{v_1}{\left|v\right|}\\ \Rightarrow \tan \theta =\frac{v_2}{v_1}\end{array}$

Know that the magnitude of a vector $r=v_1,v_2$ is equal to:

  $\left|v\right|=\sqrt{v_1^2+v_2^2}$

Let, the mark:

  $u=(\sqrt{3},0)$

Then, the magnitude of a vector:

  $\begin{array}{c}\left|u\right|=\sqrt{{(\sqrt{3})}^2+0^2}\\ =\sqrt{3+0}\\ =\sqrt{3}.\end{array}$

To use the previous result,

  $\begin{array}{c}\tan {\theta }_u=\frac{0}{\sqrt{3}}\\ =0\\ \Rightarrow {\theta }_u=0^{\circ }\end{array}$

Let the mark,

  $v=(0,1)$

Then, the magnitude of a vector:

  $\begin{array}{c}\left|v\right|=\sqrt{0^2+1^2}\\ =\sqrt{1}\\ =1\end{array}$

To use the previous result,

  $\begin{array}{c}\tan {\theta }_v=\frac{1}{0}\\ =\infty \\ \Rightarrow {\theta }_v={90}^{\circ }\end{array}$

According to the previous result,

  $\begin{array}{c}u+v=(\sqrt{3},0)+(0,1)\\ =(\sqrt{3},1)\end{array}$

Then, the magnitude of the vector:

  $\begin{array}{c}|u+v|=\sqrt{{(\sqrt{3})}^2+1^2}\\ =\sqrt{3+1}\\ =\sqrt{4}\\ =2\end{array}$

To use the previous result,

  $\begin{array}{c}\tan \theta =\frac{1}{\sqrt{3}}\\ \Rightarrow \theta ={30}^{\circ }\end{array}$

Because,

  $\begin{array}{c}\sin {\theta }_v=1/2>0\\ \cos {\theta }_v=\sqrt{3}/2>0\end{array}$

According to the previous results, conclude that for the vector $u=(\sqrt{3},0)$ which has the direction angle ${\theta }_u=0^{\circ }$ for the vector

  $v=(0,1)$ which has the direction angle

  ${\theta }_v={90}^{\circ }$ .The direction angle of

the vector $u+v$ is equal to ${30}^{\circ }$ .

Therefore, the required given statement is false.