Question

Given \[\vec P = P\cos \theta \hat i + P\sin \theta \hat j\] . The vector \[\vec P\]which is perpendicular to \[\vec Q\] is given by
(A) \[Q\cos \theta \hat i - Q\sin \theta \hat j\]
(B) \[Q\cos \theta \hat i - Q\sin \theta \hat j\]
(C) \[Q\cos \theta \hat i + Q\sin \theta \hat j\]
(D) \[Q\sin \theta \hat i + Q\cos \theta \hat j\]

Answer 1

Hint:The above problem can be resolved by using the concepts of the vector dot product. The vector is that physical quantity, which is defined in terms of direction only and no need of the magnitude, is there. The vector quantity has several applications along with the fact that, if any mathematical equation is given for two vectors, then it is resolved by taking the dot product of the vectors. Moreover, the condition of the perpendicularity of two vectors is given when the two vectors' dot product is zero.

Complete step by step answer:
As the vector P and the vector Q are perpendicular to each other, this makes the dot product of the vector to be zero.
Select the appropriate value of vector Q from the given option as,
\[\begin{array}{l}
\vec P = P\cos \theta \hat i + P\sin \theta \hat j\\
\vec P \cdot \vec Q = \left( {P\cos \theta \hat i + P\sin \theta \hat j} \right)\left( {Q\cos \theta \hat i - Q\sin \theta \hat j} \right)\\
\vec P \cdot \vec Q = \left( {P\cos \theta \hat i + P\sin \theta \hat j} \right)\left( {Q\cos \theta \hat i - Q\sin \theta \hat j} \right)\\
\vec P \cdot \vec Q = - PQ\sin \theta \cos \theta + PQ\sin \theta \cos \theta \\
\vec P \cdot \vec Q = 0
\end{array}\]
Therefore, the vectors P and Q are perpendicular to each other, when the magnitude of vector Q is \[Q\cos \theta \hat i - Q\sin \theta \hat j\] and the option (B ) is correct.

Note: Try to understand the concept of vectors and the actual application of vectors. The vector is the type of physical quantity used to define the direction of any variable. Moreover, the vector has its more comprehensive applications in modern physics and Mathematics calculations.