Chapter 3: Q. 17 (page 78)

Let $\vec{E} = 2 \hat{i} + 3 \hat{j} a n d \vec{F} = 2 \hat{i} - 2 \hat{j} .$ Find the magnitudes of
a. $\vec{E} a n d \vec{F}$ b. $\vec{E} + \vec{F}$ c. $- \vec{E} - 2 \vec{F}$

Short Answer

Expert verified

(a) The magnitude of $\vec{E}$ is $\sqrt{13}$ and $\vec{F}$ is $\sqrt{8} .$

(b) The magnitude of $\vec{E} + \vec{F}$ is $\sqrt{17} .$

Step by step solution

Part (a) Step 1. Given information

It is given that $\vec{E} = 2 \hat{i} + 3 \hat{j} a n d \vec{F} = 2 \hat{i} - 2 \hat{j} .$ We need to determine the magnitude of $E$ and $F .$

Step 2. Calculation

$\vec{E} = 2 \hat{i} + 3 \hat{j} .$

$E_{x} = 2 a n d E_{y} = 3 .$

$|\vec{E}| = \sqrt{{(E_{x})}^{2} + {(E_{y})}^{2}} .$

$= \sqrt{{(2)}^{2} + {(3)}^{2}} .$

$= \sqrt{4 + 9} .$

$= \sqrt{13} .$

$\vec{F} = 2 \hat{i} - 2 \hat{j} .$

$F_{x} = 2, F_{y} = - 2 .$

$|\vec{F}| = \sqrt{{(F_{x})}^{2} + {(F_{y})}^{2}} .$

$= \sqrt{{(2)}^{2} + {(- 2)}^{2}} .$

$= \sqrt{4 + 4} .$

$= \sqrt{8} .$

Part (b) Step 1. Given information

It is given that, $\vec{E} = 2 \hat{i} + 3 \hat{j} a n d \vec{F} = 2 \hat{i} - 2 \hat{j} .$ We need to determine the magnitude of $\vec{E} + \vec{F} .$

Step 2. Calculation

$\vec{E} = 2 \hat{i} + 3 \hat{j} a n d \vec{F} = 2 \hat{i} - 2 \hat{j} .$

$\vec{E} + \vec{F} = (2 \hat{i} + 3 \hat{j}) + (2 \hat{i} - 2 \hat{j} .)$

$= 4 \hat{i} - \hat{j} .$

role="math" localid="1649179077562" ${(E + F)}_{x} = 4, {(E + F)}_{y} = - 1 .$

$|\vec{E} + \vec{F}| = \sqrt{{(E + F)}_{x}^{2} + {(E + F)}_{y}^{2}} .$

$= \sqrt{{(4)}^{2} + {(- 1)}^{2} .}$

$= \sqrt{16 + 1} .$

$= \sqrt{17} .$

Part (c) Step 1. Given information

It is given that, $\vec{E} = 2 \hat{i} + 3 \hat{j} a n d \vec{F} = 2 \hat{i} - 2 \hat{j} .$ We need to determine the magnitude of $- \vec{E} + 2 \vec{F} .$

Step 2. Calculation

$\vec{E} = 2 \hat{i} + 3 \hat{j} a n d \vec{F} = 2 \hat{i} - 2 \hat{j} .$

$- \vec{E} - 2 \vec{F} .$

$= - (2 \hat{i} + 3 \hat{j}) - 2 (2 \hat{i} - 2 \hat{j}) .$

$= - 2 \hat{i} - 3 \hat{j} - 4 \hat{i} + 4 \hat{j} .$

$= - 6 \hat{i} + \hat{j} .$

${(- E + 2 F)}_{x} = - 6, {(- E + 2 F)}_{y} = 1 .$

$|- \vec{E} + 2 \vec{F}| = \sqrt{{(- E + 2 F)}^{2}_{x} + {(- E + 2 F)}^{2}_{y}} .$

$= \sqrt{{(- 6)}^{2} + {(1)}^{2} .}$

$= \sqrt{36 + 1} .$

$= \sqrt{37} .$

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Let $\vec{E} = 2 \hat{i} + 3 \hat{j} a n d \vec{F} = 2 \hat{i} - 2 \hat{j} .$ Find the magnitudes of
a. $\vec{E} a n d \vec{F}$ b. $\vec{E} + \vec{F}$ c. $- \vec{E} - 2 \vec{F}$

Short Answer

Step by step solution

Part (a) Step 1. Given information

Step 2. Calculation

Part (b) Step 1. Given information

Step 2. Calculation

Part (c) Step 1. Given information

Step 2. Calculation

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Let E→=2i^+3j^andF→=2i^-2j^.Find the magnitudes of a.E→andF→b.E→+F→c.-E→-2F→

Short Answer

Step by step solution

Part (a) Step 1. Given information

Step 2. Calculation

Part (b) Step 1. Given information

Step 2. Calculation

Part (c) Step 1. Given information

Step 2. Calculation

One App. One Place for Learning.

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Let $\vec{E} = 2 \hat{i} + 3 \hat{j} a n d \vec{F} = 2 \hat{i} - 2 \hat{j} .$ Find the magnitudes of
a. $\vec{E} a n d \vec{F}$ b. $\vec{E} + \vec{F}$ c. $- \vec{E} - 2 \vec{F}$