Q64 / 120 · JEE Main 2007 · Mathematics

If $\hat{u}$ and $\hat{v}$ are unit vectors and $\theta$ is the acute angle between them, then $2 \hat{u} \times 3 \hat{v}$ is a unit vector for

  • A
    exactly two values of $\theta$
  • B
    more than two values of $\theta$
  • C
    no value of $\theta$
  • Correct answer
    D
    exactly one value of $\theta$

Solution

$|2 \hat{u} \times 3 \hat{v}|=1$ $\begin{aligned} & 6|\hat{u}\|\hat{v} \| \sin \theta|=1 \\ & \sin \theta=\frac{1}{6} \end{aligned}$ Hence there is exactly one value of $\theta$ for which $2 \hat{u} \times 3 \hat{v}$ is a unit vector.

Hence, the correct answer is option: 'D'.

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