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Given: P = A + B and Q = A – B. If the magnitudes of vectors P and Q are equal, what is the angle between A and B?

\[A.\,\text{zero}\]
\[B.\,\dfrac{\pi }{4}\]
\[C.\dfrac{\pi }{2}\]
\[D.\,\pi \]

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Last updated date: 20th Jun 2024
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Answer
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Hint: The given question is based on the angle between the vectors, so, we will make use of the properties of vectors. Firstly, we will find the magnitudes of the vectors to get an equation in terms of the angle. Then, we will equate the magnitudes of these vectors to obtain the value of the angle between the vectors.

Complete answer:
From given, we have the vectors,
P = A + B and Q = A – B
Firstly, compute the magnitudes of the given vectors. So, we have,
\[|P|=|A+B|\] and \[|Q|=|A-B|\]
Upon further calculation, we get,
\[|P{{|}^{2}}={{A}^{2}}+{{B}^{2}}+2AB\cos \theta \] and \[|Q{{|}^{2}}={{A}^{2}}+{{B}^{2}}-2AB\cos \theta \]
Now equate the magnitudes of these vectors. So, we get,
\[|P{{|}^{2}}=|Q{{|}^{2}}\]
Substitute the values of the magnitudes.
\[{{A}^{2}}+{{B}^{2}}+2AB\cos \theta ={{A}^{2}}+{{B}^{2}}-2AB\cos \theta \]
Cancel out the common terms.
\[\begin{align}
  & 2AB\cos \theta +2AB\cos \theta =0 \\
 & \Rightarrow 4AB\cos \theta =0 \\
\end{align}\]
Compute the value of the angle.
\[\begin{align}
  & \cos \theta =\dfrac{0}{4AB} \\
 & \Rightarrow \cos \theta =0 \\
\end{align}\]
Now rearrange the above equation.
\[\begin{align}
  & \theta ={{\cos }^{-1}}(0) \\
 & \Rightarrow \theta =\dfrac{\pi }{2} \\
\end{align}\]
This value of the angle between the vectors describes that the vectors are perpendicular to each other.
The angle between the vectors, in turn, defines the direction of the vectors. For example, if the angle between the vectors is \[90{}^\circ \], the direction of vectors are perpendicular to each other, if the angle between the vectors is \[0{}^\circ \], the direction of vectors are parallel to each other and if the angle between the vectors is \[180{}^\circ \], the direction of vectors are opposite to each other, both forming a straight line.
The value of the angle between the given vectors is \[\dfrac{\pi }{2}\].

So, the correct answer is “Option C”.

Note:
In this case, we are asked to find the value of the angle between the vectors. So, we have involved the cos function. This value of angle obtained is 90 degrees. If in some cases, we obtain the value of a given angle to be equal to zero, then, the vectors are considered to be parallel to each other.