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The resultant of two vectors acting at an angle ${{150}^{\circ }}$ is of magnitude 10 units and is perpendicular to one of the vectors. The magnitude of the other vector is:
A. $\dfrac{20}{\sqrt{3}}$ N
B. $10\sqrt{3}$ N
C. 20 N
D. $20\sqrt{3}$ N

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Last updated date: 13th Jun 2024
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Answer
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Hint: First draw the diagram as per the question. Now we will see which law can we apply and how to place the vectors to satisfy the law that we have chosen. Now apply the law and form an equation, now we have to solve the equation to get the answer.

Complete step by step answer:
The question says that the resultant has a magnitude of 10 units and it is perpendicular to one of the vectors.
And the resultant is at an angle of ${{150}^{\circ }}$.
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In the above diagram we see a vector A and another vector B which has an angle of a=${{150}^{\circ }}$, between them, now we are asked to find the resultant vector that is shown as ‘R’ and to do that we are changing the position on B vector from point d to point e.
Now, we know that the total angle between A and B vector was ${{150}^{\circ }}$, now we know that the resultant is making an angle of ${{90}^{\circ }}$ with A vector.
Now, as angle a is ${{150}^{\circ }}$then angle ‘b’ must also be same as the vector is just shifted and no angle was changed.
So, we can calculate angle ‘c’ by subtracting ${{180}^{\circ }}$from ${{150}^{\circ }}$as the line is a straight line,
so the angle will be,
$c={{150}^{\circ }}-{{130}^{\circ }}={{30}^{\circ }}$
Now, we have to find the magnitude of vector A as it has the smallest force,
To calculate vector A,
We will apply triangle law,
So, we know that the formula is,
$\tan \theta =\dfrac{perpendicular}{Base}$ ,
Now placing the values,
$\tan c=\dfrac{|R|}{|A|}$,
$\tan {{30}^{\circ }}=\dfrac{10}{|A|}$,
$\dfrac{1}{\sqrt{3}}=\dfrac{10}{|A|}$(we know that according to trigonometrical table, $\tan {{30}^{\circ }}=\dfrac{1}{\sqrt{3}}$)
Therefore on solving this we will get the magnitude of vector A as,
$|A|=10\sqrt{3}$,
So, the correct option is option B.

Note: While drawing the diagram we must carefully look into the question and draw the diagram as if the diagram is wrong then the whole question will go wrong. Now find the angles correctly because the angles matter in the equation. Remember the triangle law to solve the answer. Students must also remember all the values in the trigonometrical tables.