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Found in: Page 56

### Fundamentals Of Physics

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

# ${\mathbf{If}}{\mathbf{}}\stackrel{\mathbf{⇀}}{\mathbf{d}}{\mathbf{=}}\stackrel{\mathbf{⇀}}{\mathbf{a}}{\mathbf{+}}\stackrel{\mathbf{⇀}}{\mathbf{b}}{\mathbf{+}}\left(-\stackrel{⇀}{c}\right){\mathbf{}}{\mathbf{,}}{\mathbf{}}{\mathbf{does}}{\mathbf{}}\left(a\right){\mathbf{}}\stackrel{\mathbf{⇀}}{\mathbf{a}}{\mathbf{+}}\left(-\stackrel{⇀}{d}\right){\mathbf{=}}\stackrel{\mathbf{⇀}}{\mathbf{c}}{\mathbf{+}}\left(-\stackrel{⇀}{b}\right){\mathbf{}}{\mathbf{,}}{\mathbf{}}\left(b\right){\mathbf{}}{\mathbf{}}\stackrel{\mathbf{⇀}}{\mathbf{a}}{\mathbf{=}}\left(-\stackrel{⇀}{b}\right){\mathbf{+}}\stackrel{\mathbf{⇀}}{\mathbf{d}}{\mathbf{+}}\stackrel{\mathbf{⇀}}{\mathbf{c}}\phantom{\rule{0ex}{0ex}}\stackrel{\mathbf{⇀}}{\mathbf{c}}{\mathbf{+}}\left(-\stackrel{⇀}{d}\right){\mathbf{=}}\stackrel{\mathbf{⇀}}{\mathbf{a}}{\mathbf{+}}\stackrel{\mathbf{⇀}}{\mathbf{b}}{\mathbf{?}}$

$\left(\mathrm{a}\right)\stackrel{⇀}{a}+\left(-\stackrel{⇀}{d}\right)=\stackrel{⇀}{c}+\left(-\stackrel{⇀}{b}\right)\mathrm{is}\mathrm{valid}\phantom{\rule{0ex}{0ex}}\left(\mathrm{b}\right)\stackrel{⇀}{\mathrm{a}}=\left(-\stackrel{⇀}{\mathrm{b}}\right)+\stackrel{⇀}{\mathrm{d}}+\stackrel{⇀}{\mathrm{c}}\mathrm{is}\mathrm{valid}\phantom{\rule{0ex}{0ex}}\left(\mathrm{c}\right)\stackrel{⇀}{\mathrm{c}}+\left(-\stackrel{⇀}{\mathrm{d}}\right)=\stackrel{⇀}{\mathrm{a}}+\stackrel{⇀}{\mathrm{b}}\mathrm{is}\mathrm{not}\mathrm{valid}$

See the step by step solution

## Step 1: Given information

The vector is given by,

$\stackrel{⇀}{d}=\stackrel{⇀}{a}+\stackrel{⇀}{b}+\left(-\stackrel{⇀}{c}\right)$

## Step 2: To understand the concept

The vector operations are different than the scalar operations. We cannot add or subtract the vectors like scalars. The vector operations are governed by the set of laws for vector addition and vector subtraction.

This problem involves a basic substation operation in which the given vector equation $\stackrel{⇀}{d}=\stackrel{⇀}{a}+\stackrel{⇀}{b}+\left(-\stackrel{⇀}{c}\right)$ can be used in each equation and checked on both sides of the equation.

Formula:

$\stackrel{⇀}{a}+\stackrel{⇀}{b}=\stackrel{⇀}{c}$

## Step 3: (a) To check validity for a⇀+(-d⇀)=c⇀+(-b⇀)

$\stackrel{⇀}{a}+\left(\stackrel{⇀}{a}-\stackrel{⇀}{b}+\stackrel{⇀}{c}\right)=\stackrel{⇀}{c}+\left(-\stackrel{⇀}{b}\right)\phantom{\rule{0ex}{0ex}}\left(-\stackrel{⇀}{b}\right)+\stackrel{⇀}{c}=\stackrel{⇀}{c}+\left(-\stackrel{⇀}{b}\right)$

As $\left(-\stackrel{⇀}{b}\right)$ and are commutative, so both sides are equal.

Hence, this is a correct equation.

## Step 4: (b) To check validity for a⇀=(-b⇀)+d⇀+c⇀

$\stackrel{⇀}{a}=\left(-\stackrel{⇀}{b}\right)+\stackrel{⇀}{a}+\stackrel{⇀}{b}+\left(-\stackrel{⇀}{c}\right)+\stackrel{⇀}{c}\phantom{\rule{0ex}{0ex}}\stackrel{⇀}{a}=\stackrel{⇀}{a}$.

Here, both sides are equal.

So, this is a correct equation.

## Step 5: (c) To check validity for c⇀+(-d⇀)=a⇀+b⇀

$\stackrel{⇀}{c}+\left(-\stackrel{⇀}{a}-\stackrel{⇀}{b}+\stackrel{⇀}{c}\right)=\stackrel{⇀}{a}+\stackrel{⇀}{b}\phantom{\rule{0ex}{0ex}}\left(-\stackrel{⇀}{a}-\stackrel{⇀}{b}\right)+2\stackrel{⇀}{c}=\stackrel{⇀}{a}+\stackrel{⇀}{b}$

Here, both sides are not equal.

So, this is not a correct equation.

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