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### Fundamentals Of Physics

Book edition 10th Edition
Author(s) David Halliday
Pages 1328 pages
ISBN 9781118230718

# The two vectors $\stackrel{\mathbf{\to }}{\mathbf{a}\mathbf{}}{\mathbit{a}}{\mathbit{n}}{\mathbit{d}}{\mathbf{}}\stackrel{\mathbf{\to }}{\mathbf{b}}$ in Fig.3.28 have equal magnitudes of 10.0m and the angles are ${\mathbit{\theta }}{\mathbf{=}}{\mathbf{30}}{\mathbf{°}}$and ${\mathbit{\theta }}{\mathbf{=}}{\mathbf{105}}{\mathbf{°}}$ . Find the (a) x and (b) y components of their vector sum$\stackrel{\mathbf{\to }}{\mathbf{r}}$ , (c) the magnitude of $\stackrel{\mathbf{\to }}{\mathbf{r}}$, and (d) the angle$\stackrel{\mathbf{\to }}{\mathbf{r}}$makes with the positive direction of the x axis

a) x component of vector sum $\stackrel{\to }{\mathrm{r}}{}{\mathrm{is}}{}{1}{.}{59}{}{\mathrm{m}}$

b) y component of vector sum $\stackrel{\to }{\mathrm{r}}{}{\mathrm{is}}{}{12}{.}{1}{}{\mathrm{m}}$

c) Magnitude of vector sum $\stackrel{\to }{\mathrm{r}}{}{\mathrm{is}}{}{12}{.}{2}{\mathrm{m}}$

c) Angle $\stackrel{\to }{\mathrm{r}}{}{\mathrm{is}}{}{82}{.}{5}{°}$

See the step by step solution

## Step 1: To understand the concept

This problem involves the addition of two vectors $\stackrel{\mathbf{\to }}{\mathbf{a}}$ and $\stackrel{\mathbf{\to }}{\mathbf{b}}$ . Using the following relations x and y components of the vector sum$\stackrel{\mathbf{\to }}{\mathbf{r}}$can be found and similarly the magnitude and the angle respectively.

localid="1656155400549" $r=\stackrel{\to }{\left|r\right|}=\sqrt{{\left(x\right)}^{2}+{\left(y\right)}^{2}}$ (i)

$\theta ={\mathrm{tan}}^{-1}\left(\frac{y}{x}\right)$ (ii)

Given

Two vectors and $\stackrel{\to }{\mathrm{a}}=10\mathrm{m}\mathrm{and}\stackrel{\to }{\mathrm{b}}=10\mathrm{m}$

two angles and ${\theta }_{1}=30°\mathrm{and}{\mathrm{\theta }}_{2}=105°$

## Step 2: To find x component of vector sum r→

x component of the vector sum r is given by

${r}_{x}=a\mathrm{cos}{\theta }_{1}+b\mathrm{cos}{\theta }_{2}\phantom{\rule{0ex}{0ex}}\stackrel{\to }{r}=10\mathrm{m}×\mathrm{cos}\left(30\right)+10\mathrm{m}×\left(135\right)\phantom{\rule{0ex}{0ex}}{\mathrm{r}}_{\mathrm{x}}=1.59\mathrm{m}$

## Step 3: To find y component of vector sum r→

y component of vector sum $\stackrel{\to }{r}$ is

${r}_{y}=\mathrm{a}{\mathrm{sin\theta }}_{1}+{\mathrm{bsin\theta }}_{2}\phantom{\rule{0ex}{0ex}}\stackrel{\to }{\mathrm{r}}=10\mathrm{m}×\mathrm{sin}\left(30\right)+10\mathrm{m}×\mathrm{sin}\left(135\right)\phantom{\rule{0ex}{0ex}}{\mathrm{r}}_{\mathrm{y}}=12.1\mathrm{m}$

## Step 4: To find magnitude of vector sum r→

Using equation (i) the magnitude of vector$\stackrel{\to }{r}$is written as

$r=\stackrel{\to }{\left|r\right|}=\sqrt{\left(1.59{\mathrm{m}}^{2}\right)+{\left(12.1\mathrm{m}\right)}^{2}}\phantom{\rule{0ex}{0ex}}r=12.2\mathrm{m}$

## Step 5: To find angle of vector sum r→

The angle $\stackrel{\to }{r}$ makes with the positive direction of the x axis is

${\mathrm{tan}}^{1}\left(\frac{12.1}{1.59}\right)=82.5°$

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