Question

a. What is the angle $\varphi$between vectors $\overrightarrow{E}and\overrightarrow{F}$in FIGURE P3.24?

b. Use geometry and trigonometry to determine the magnitude and direction of $\overrightarrow{G}=\overrightarrow{E}+\overrightarrow{F}.$

c. Use components to determine the magnitude and direction of $\overrightarrow{G}=\overrightarrow{E}+\overrightarrow{F}.$

Short answer

(a) $\varphi =71.6^{\circ }.$

(b) The magnitude is $3$and the direction is ${45}^{\circ }.$

(c) The magnitude is$3$and the direction is${90}^{\circ }.$

Step-by-step solution

Part (a) Step 1. Given information

The given figure is,

We need to determine the magnitude and direction of$\overrightarrow{G}=\overrightarrow{E}+\overrightarrow{F}$by using geometry and trigometry.

Step 2. Calculation

The magnitude and the direction of $\overrightarrow{G}$can be determined by the cosine rule.

${\left|\overrightarrow{G}\right|}^2=2+5-2\sqrt{2}\sqrt{5}\cos \left(180-\varphi \right).$

$=2+5-2\sqrt{1}\sqrt{5}\cos \left(180-71.6\right).$

$\approx 9\Rightarrow \left|\overrightarrow{G}\right|\approx 3.$

The direction can be determined by using an angle between $G$and $E.$

$\cos \alpha =\frac{3^2+2-5}{2\left(3\right)\sqrt{2}}.$

$\cos \alpha =\frac{1}{\sqrt{2}}.$

$\alpha ={\cos }^{-1}\left(\frac{1}{\sqrt{2}}\right).$

$={45}^{\circ }.$$={45}^{\circ }.$

Part (c) Step 1. Given information

The given figure is,

We need to determine the magnitude and direction of$\overrightarrow{G}=\overrightarrow{E}+\overrightarrow{F}$by using the components.

Step 2. Calculation

$\overrightarrow{G}=\overrightarrow{E}+\overrightarrow{F}.$

role="math" $=\left[1+\left(-1\right)\right]\hat{i}+\left(1+2\right)\hat{j}.$

$=3\hat{j}.$

$\left|\overrightarrow{G}\right|=\sqrt{0^2+3^2.}$

$=\sqrt{9}.$

role="math" localid="1649272859822" $\varphi =3.$

$\tan \theta =\frac{G_y}{G_x}.$

role="math" $=\frac{3}{0}.$

$\theta ={\tan }^{-1}\left(0\right).$

$={90}^{\circ }.$