MHT CET · Physics · Mathematics in Physics
Let \(\overrightarrow{\mathrm{P}}=\hat{\mathrm{I}} \mathrm{P} \sin \theta-\hat{\mathrm{P}} \cos \theta\), be any vector. Another vector \(\overrightarrow{\mathrm{Q}}\) which is perpendicular
to \(\overrightarrow{\mathrm{P}}\) is
- A \((\hat{\mathrm{I}} \mathrm{Q} \sin \theta+\hat{\mathrm{j}} \mathrm{Q} \cos \theta)\)
- B \((\hat{\mathrm{I}} \mathrm{Q} \cos \theta+\hat{\mathrm{j}} \mathrm{Q} \sin \theta)\)
- C \((\hat{\mathrm{I}} \mathrm{Q} \cos \theta-\hat{\mathrm{j}} \mathrm{Q} \sin \theta)\)
- D \((\hat{\mathrm{l}} \mathrm{P} \sin \theta+\hat{\mathrm{j}} \mathrm{P} \cos \theta)\)
Answer & Solution
Correct Answer
(B) \((\hat{\mathrm{I}} \mathrm{Q} \cos \theta+\hat{\mathrm{j}} \mathrm{Q} \sin \theta)\)
Step-by-step Solution
Detailed explanation
\(\vec{P}=P \sin \theta \hat{i}-P \cos \theta \hat{j}\)
also, \(\vec{Q}=Q \cos \theta\hat{i}+Q \sin \theta\hat{j}\)
\(\therefore \vec{P} .\vec{Q}=(P \sin \theta \hat{i}-P \cos \theta \hat{j}) \cdot(Q \cos \theta \hat{i}+Qsin\theta \hat{j})\)
\(=P Q \sin \theta \cos \theta-P \cos \theta- Q\sin \theta\)
\(=0\)
also, \(\vec{Q}=Q \cos \theta\hat{i}+Q \sin \theta\hat{j}\)
\(\therefore \vec{P} .\vec{Q}=(P \sin \theta \hat{i}-P \cos \theta \hat{j}) \cdot(Q \cos \theta \hat{i}+Qsin\theta \hat{j})\)
\(=P Q \sin \theta \cos \theta-P \cos \theta- Q\sin \theta\)
\(=0\)
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