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Karnataka Board PUCPUC Science Class 11

Add Vectors → a , → B and → C Each Having Magnitude of 100 Unit and Inclined to the X-axis at Angles 45°, 135° and 315° Respectively.

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Question

Add vectors \[\vec{A} , \vec{B} \text { and } \vec{C}\]  each having magnitude of 100 unit and inclined to the X-axis at angles 45°, 135° and 315° respectively.

Answer in Brief
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Solution

First, we will find the components of the vector along the x-axis and y-axis. Then we will find the resultant x and y-components.  
x-component of \[\vec{A} = \ A\ cos \ 45^\circ =100 \cos 45^\circ = \frac{100}{\sqrt{2}} \text { unit }\]

x-component of \[\vec{B} = \vec{B} \cos 135^\circ = - \frac{100}{\sqrt{2}}\]

x-component of \[\vec{C}\] = \[\vec{C}\] cos 315\[^\circ\]

= 100 cos 315°

\[= 100 \cos 45^\circ = \frac{100}{\sqrt{2}}\]
Resultant x-component \[= \frac{100}{\sqrt{2}} - \frac{100}{\sqrt{2}} + \frac{100}{\sqrt{2}} = \frac{100}{\sqrt{2}}\]
Now, y-component of \[\vec{A} = 100 \sin 45^\circ = \frac{100}{\sqrt{2}}\]
y-component of \[\vec{B} = 100 \sin 135^\circ = \frac{100}{\sqrt{2}}\] 
y-component of \[\vec{C} = 100 \sin 315^\circ = - \frac{100}{\sqrt{2}}\]
Resultant y-component
\[= \frac{100}{\sqrt{2}} + \frac{100}{\sqrt{2}} - \frac{100}{\sqrt{2}} = \frac{100}{\sqrt{2}}\]
Magnitude of the resultant \[= \sqrt{\left( \frac{100}{\sqrt{2}} \right)^2 + \left( \frac{100}{\sqrt{2}} \right)^2}\] 
 
\[= \sqrt{10000} = 100\]
Angle made by the resultant vector with the x-axis is given by

\[\tan \alpha = \frac{\text { y comp}}{\text { x comp }}\]

\[ = \frac{100\sqrt{2}}{100\sqrt{2}} = 1\]

⇒ α = tan−1 (1) = 45°

∴ The magnitude of the resultant vector is 100 units and it makes an angle of 45° with the x-axis.

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Chapter 2: Physics and Mathematics - Exercise [Page 29]

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HC Verma Concepts of Physics Volume 1 and 2 [English]
Chapter 2 Physics and Mathematics
Exercise | Q 3 | Page 29

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