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Q22 P

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Found in: Page 73

### Physics For Scientists & Engineers

Book edition 9th Edition
Author(s) Raymond A. Serway, John W. Jewett
Pages 1624 pages
ISBN 9781133947271

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# Use the component method to add the $\stackrel{\to }{A}$ vectors and $\stackrel{\to }{B}$ shown in Figure P3.11. Both vectors have magnitudes of $3.00m$ and vector $\stackrel{\to }{A}$ makes an angle of with the x axis. Express the resultant $\stackrel{\to }{\mathrm{A}}+\stackrel{\to }{\mathrm{B}}$ in unit-vector notation.

$\stackrel{\to }{\mathrm{A}}+\stackrel{\to }{\mathrm{B}}=\left(2.6\stackrel{^}{\mathrm{i}}+4.5\stackrel{^}{\mathrm{j}}\right)\mathrm{m}$

See the step by step solution

## Step 1: Define the components

In a two-dimensional coordinate system, the x-component and y-component are commonly considered to be the components of a vector. It can be written as $V\text{}=\text{}\left(vx,\text{}vy\right)$ with V denoting the vector.

These are the components of vectors created along the axes. In this article, we shall find the components of any given vector using formulas for both two-dimensional and three-dimensional coordinate systems.

${v}_{x}=V\mathrm{cos}\theta \phantom{\rule{0ex}{0ex}}{v}_{y}=V\mathrm{sin}\theta$

## Step 2: State the given data

$A=3\text{m}\phantom{\rule{0ex}{0ex}}B=3\text{m}\phantom{\rule{0ex}{0ex}}{\theta }_{A}=30°\phantom{\rule{0ex}{0ex}}{\theta }_{B}=90°$

## Step 3: Find the component vector A

Discover the following components to express A in component form:

${A}_{x}=A\mathrm{cos}{\theta }_{A}\phantom{\rule{0ex}{0ex}}=3\mathrm{cos}\left(30°\right)\phantom{\rule{0ex}{0ex}}=2.6m$

${A}_{y}=A\mathrm{cos}{\theta }_{A}\phantom{\rule{0ex}{0ex}}=3\mathrm{cos}\left(30°\right)\phantom{\rule{0ex}{0ex}}=1.5m$

$\stackrel{\to }{\mathrm{A}}={\mathrm{A}}_{\mathrm{x}}\stackrel{^}{\mathrm{i}}+{\mathrm{A}}_{\mathrm{y}}\stackrel{^}{\mathrm{j}}\phantom{\rule{0ex}{0ex}}=\left(2.6\stackrel{^}{\mathrm{i}}+1.5\stackrel{^}{\mathrm{j}}\right)\mathrm{m}$

## Step 4: Find the component vector B

Discover the following components to express B in component form

${B}_{x}=0\text{m}\phantom{\rule{0ex}{0ex}}{B}_{y}=3\text{m}\phantom{\rule{0ex}{0ex}}\stackrel{\to }{\mathrm{B}}={\mathrm{B}}_{\mathrm{x}}\stackrel{^}{\mathrm{i}}+{\mathrm{B}}_{\mathrm{y}}\stackrel{^}{\mathrm{j}}\phantom{\rule{0ex}{0ex}}=\left(3\stackrel{^}{\mathrm{j}}\right)\mathrm{m}\phantom{\rule{0ex}{0ex}}$

Now find the resultant of A and B. We will add the components.

$\stackrel{\to }{\mathrm{A}}+\stackrel{\to }{\mathrm{B}}=\left(2.6\stackrel{^}{\mathrm{i}}+1.5\stackrel{^}{\mathrm{j}}\right)\mathrm{m}+\left(3\stackrel{^}{\mathrm{j}}\right)\mathrm{m}\phantom{\rule{0ex}{0ex}}=\left(2.6\stackrel{^}{\mathrm{i}}+4.5\stackrel{^}{\mathrm{j}}\right)\mathrm{m}$

The resultant vector is $\stackrel{\to }{\mathrm{A}}+\stackrel{\to }{\mathrm{B}}=\left(2.6\stackrel{^}{\mathrm{i}}+4.5\stackrel{^}{\mathrm{j}}\right)\mathrm{m}$

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