A) \[b\,\sin \,\theta \,\hat{i}-a\,\cos \,\theta \,\hat{j}\]
B) \[\frac{1}{a}\,\sin \,\theta \,\hat{i}-a\,\cos \,\theta \,\hat{j}\]
C) \[5\hat{k}\]
D) All of the above
Correct Answer: D
Solution :
From definition of dot product of vectors, we have \[x\,.\,y=xy\,\,\cos \,\,\theta \] When \[\theta ={{90}^{o}},\,\,\,\cos \,\,{{90}^{o}}=0\] \[\therefore \] \[x\,.\,y=0\] Given, \[x=a\,\cos \,\theta \,\hat{i}+b\,\sin \,\theta \,\hat{j}\] \[y=b\,sin\,\,\theta \,\,\hat{i}-a\,\cos \,\,\theta \,\hat{j}\] \[x\,.\,y=(a\,\cos \,\,\theta \,\,\hat{i}+b\,sin\,\,\theta \,\hat{j})\] \[(b\,sin\,\,\theta \,\,\hat{i}-a\,\cos \,\,\theta \,\hat{j})\] \[x\,.\,y=ab\,\,\sin \,\theta \,\,\cos \,\theta \,-\,ab\,\,\,\sin \,\theta \,\,\cos \,\theta \,=0\] Hence, vectors are perpendicular. Similarly for option and also \[x.y=0\]
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