Chapter 3: Q. 16 (page 78)

Let $\vec{A} = 4 \hat{i} - 2 \hat{j}, \vec{B} = - 3 \hat{i} + 5 \hat{j} a n d \vec{F} = \vec{A} - 4 \vec{B} .$
a. Write vector $\vec{F}$ in component form.
b. Draw a coordinate system and on it show vectors $\vec{A}, \vec{B} a n d \vec{F} .$
c. What are the magnitude and direction of vector $\vec{F} ?$

Short Answer

Expert verified

(a) $\vec{F} = 20 \hat{i} - 22 \hat{j} .$

(b) The graph is,

Step by step solution

Part (a) Step 1. Given information

It is given that, $\vec{A} = 4 \hat{i} - 2 \hat{j}, \vec{B} = - 3 \hat{i} + 5 \hat{j} a n d \vec{F} = \vec{A} - 4 \vec{B} .$ We need to write the vector $\vec{F}$ in component form.

Step 2. Solution

$\vec{A} = 4 \hat{i} - 2 \hat{j}, \vec{B} = - 3 \hat{i} + 5 \hat{j} .$

$\vec{F} = \vec{A} - 4 \vec{B} .$

$= (4 \hat{i} - 2 \hat{j}) - 4 (- 3 \hat{i} + 5 \hat{j}) .$

$= 4 \hat{i} - 2 \hat{j} + 12 \hat{i} - 20 \hat{j} .$

$= 20 \hat{i} - 22 \hat{j} .$

Part (b) Step 1. Given information

It is given that $\vec{A} = 4 \hat{i} - 2 \hat{j}, \vec{B} = - 3 \hat{i} + 5 \hat{j} a n d \vec{F} = \vec{A} - 4 \vec{B} = 20 \hat{i} - 22 \hat{j} .$ We need to draw the coordinate system and on it show vector $\vec{A}, \vec{B} a n d \vec{F} .$

Step 2. Graph

The graph of $\vec{A} = 4 \hat{i} - 2 \hat{j}, \vec{B} = - 3 \hat{i} + 5 \hat{j} a n d \vec{F} = 20 \hat{i} - 22 \hat{j}$ is,

Part (c) Step 1. Given information

It is given that $\vec{A} = 4 \hat{i} - 2 \hat{j}, \vec{B} = - 3 \hat{i} + 5 \hat{j} a n d \vec{F} = \vec{A} - 4 \vec{B} = 20 \hat{i} - 22 \hat{j} .$ We need to determine the magnitude and direction of $\vec{F} .$

Step 2. Calculation

$\vec{F} = 20 \hat{i} - 22 \hat{j} .$

$F_{x} = 20 a n d F_{y} = - 22 .$

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

$= \sqrt{{(20)}^{2} + {(- 22)}^{2} .}$

$= \sqrt{400 + 484} .$

$= \sqrt{884} .$

$= 2 \sqrt{221} .$

$\tan θ = \frac{F_{y}}{F_{x}} = \frac{- 22}{20} = - \frac{11}{10} .$

$θ = \tan^{- 1} (\frac{- 11}{10}) .$

$= - 47 . 7^{\circ} .$

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Short Answer

Step by step solution

Part (a) Step 1. Given information

Step 2. Solution

Part (b) Step 1. Given information

Step 2. Graph

Part (c) Step 1. Given information

Step 2. Calculation

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Let A→=4i^-2j^,B→=-3i^+5j^andF→=A→-4B→.a. Write vector F→in component form.b. Draw a coordinate system and on it show vectors A→,B→andF→.c. What are the magnitude and direction of vectorF→?

Short Answer

Step by step solution

Part (a) Step 1. Given information

Step 2. Solution

Part (b) Step 1. Given information

Step 2. Graph

Part (c) Step 1. Given information

Step 2. Calculation

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Most popular questions from this chapter

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