Book edition 10th Edition

Author(s) David Halliday

Pages 1328 pages

ISBN 9781118230718

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Short Answer

Three vectors $\vec{a}, \vec{b} a n d \vec{c}$ each have a magnitude of 50m and lie in an xy plane. Their directions relative to the positive direction of the x axis are $30 °$ , $195 °$ , and $315 °$ , respectively. What are (a) the magnitude and (b) the angle of the vector localid="1656259759790" $\vec{a} + \vec{b} + \vec{c}$ , and (c) the magnitude and (d) the angle of $\vec{a} + \vec{b} + \vec{c}$ ? What are the (e) magnitude and (f) angle of a fourth vector $\vec{d}$ such that $(\vec{a} + \vec{b}) - (\vec{c} + \vec{d}) = 0$ ?

Magnitude of vector $\vec{a} + \vec{b} + \vec{c}$ is

Angle of vector $\vec{a} + \vec{b} + \vec{c}$ is

Magnitude of vector $\vec{a} + \vec{b} + \vec{c}$ is

Angle of vector $\vec{a} + \vec{b} + \vec{c}$ is

Magnitude of fourth vector is

Angle of fourth vector is

See the step by step solution

Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: To understand the concept

Here, vector law of addition and subtraction is used to find the resultant of the given vector. Further using the general formula for the magnitude and the angle, the magnitude and the angle of the given vector can be calculated.

Formulae

$\vec{a} + \vec{b} + \vec{c} = (a_{x}) \overset{⏜}{i} + (a_{y}) \overset{⏜}{j} + (b_{x}) \overset{⏜}{i} + (b_{y}) \overset{⏜}{j} + (c_{x}) \overset{⏜}{i} + (c_{y}) \overset{⏜}{j} \vec{a} + \vec{b} + \vec{c} = (a_{x} + b_{x} + c_{x}) \overset{⏜}{i} + (a_{y} + b_{y} + c_{y}) \overset{⏜}{j} \vec{a} + \vec{b} = (a_{x} + b_{x}) \overset{⏜}{i} + (a_{y} + b_{y}) \overset{⏜}{j} r = |\vec{a} + \vec{b} + \vec{c}| = \sqrt{{(a_{x} + b_{x} + c_{x})}^{2} + {(a_{y} + b_{y} + c_{y})}^{2}} θ = \tan^{- 1} (\frac{a_{x} + b_{x} + c_{x}}{a_{y} + b_{y} + c_{y}}) Given are \vec{a} = (50 m) \cos (30) \overset{⏜}{i} + (50 m) \sin (30) \overset{⏜}{j} \vec{b} = (50 m) \cos (195) \overset{⏜}{i} + (50 m) \sin (195) \overset{⏜}{j} \vec{c} = (50 m) \cos (315) \overset{⏜}{i} + (50 m) \sin (315) \overset{⏜}{j}$

Step 2: To find magnitude of vector a→+b→+c→

Using the above values the vector $\vec{a} + \vec{b} + \vec{c}$ can be written as

$\vec{a} + \vec{b} + \vec{c} = (30.4) \overset{⏜}{i} (- 23.3 m) \overset{⏜}{j} Magnitude of \vec{a} + \vec{b} + \vec{c} is |\vec{a} + \vec{b} + \vec{c}| = \sqrt{(30.4 m^{2}) + {(- 23.3 m)}^{2}} |\vec{a} + \vec{b} + \vec{c}| = 38 m$

Step 3: To find the angle between vector a→+b→+c→ and x axis

The angle between $\vec{a} + \vec{b} + \vec{c}$ and x axis is

$\tan^{- 1} (\frac{- 23.3 m}{30.4 m}) = - 37.5 °$

This is equivalent to

37.5 °

clockwise from the +x axis and

322.5 °

counterclockwise from +x axis

Step 4: To find magnitude of vector a→+b→+c→

$\vec{a} + \vec{b} + \vec{c} = (127 m) \overset{⏜}{i} + (2.60 m) \overset{⏜}{j} |\vec{a} + \vec{b} + \vec{c}| = \sqrt{{(127 m)}^{2} + {(2.60 m)}^{2}} |\vec{a} + \vec{b} + \vec{c}| = 1.30 \times 10^{2} m$

Step 5: To find the angle between vector a→+b→+c→ and x axis

The angle between $\vec{a} + \vec{b} + \vec{c}$ and x axis is

$\tan^{- 1} (\frac{26.5 m}{127 m}) = 1.2 °$

Therefore, the angle between $\vec{a} + \vec{b} + \vec{c}$ and +x axis is $1.2 °$

Step 6: To find magnitude of fourth vector d→

$\vec{d} = \vec{a} + \vec{b} + \vec{c} = (- 40.4 m) \overset{⏜}{i} + (47.4 m) \overset{⏜}{j} |\vec{d}| = \sqrt{{(- 40.4 m)}^{2} + {(47.4 m)}^{2}} = 62 m |\vec{d}| = 62 m$

Step 7: To find the angle between vector d→ and x axis

The angle between $\vec{d}$ and +x axis is

$\tan^{- 1} (\frac{47.4 m}{- 40.4 m}) = 50 °$

As vector $\vec{d}$ is in third quadrant so,

$180 ° - 50 ° = 130 °$

Therefore, the angle between $\vec{d}$ and +x axis is

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Short Answer

Step by Step Solution

Step 1: To understand the concept

Step 2: To find magnitude of vector a→+b→+c→

Step 3: To find the angle between vector a→+b→+c→ and x axis

Step 4: To find magnitude of vector a→+b→+c→

Step 5: To find the angle between vector a→+b→+c→ and x axis

Step 6: To find magnitude of fourth vector d→

Step 7: To find the angle between vector d→ and x axis

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Fundamentals Of Physics

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Short Answer

Step by Step Solution

Step 1: To understand the concept

Step 2: To find magnitude of vector a→+b→+c→

Step 3: To find the angle between vector a→+b→+c→ and x axis

Step 4: To find magnitude of vector a→+b→+c→

Step 5: To find the angle between vector a→+b→+c→ and x axis

Step 6: To find magnitude of fourth vector d→

Step 7: To find the angle between vector d→ and x axis

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