Question

Vector which is perpendicular to $a\cos \theta \hat{i}+b\sin \theta \hat{j}$ is

• A. $b\sin \theta \hat{i}-a\cos \theta \hat{j}$
• B. $\frac{1}{a}\sin \theta \hat{i}-\frac{1}{b}\cos \theta \hat{j}$
• C. $5\hat{k}$
• D. All of these

Answer 1

Answer: (D)

From definition of dot product of vectors, we have
$x.y=xy\cos \theta$
When $35mm\theta =90^{\circ },cos90^{\circ }=0$
$\therefore x.y=0$
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 options (b) and (c) also x . y = 0