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

(D) exactly one value of $\theta$

(A) exactly two values of $\theta$

(B) more than two values of $\theta$

JEE Main · 2007 · 29 Apr 2007 · Offline · Vector Algebra

\overset{u}{^}

and

\overset{v}{^}

are unit vectors and

θ

is the acute angle between them, then

2 \overset{u}{^} \times 3 \overset{v}{^}

is a unit vector for

Answer: (D)

For

2 \overset{u}{^} \times 3 \overset{v}{^}

to be a unit vector its magnitude must equal one, so

6 ∣ \overset{u}{^} ∣∣ \overset{v}{^} ∣ sin θ = 1

. With

\overset{u}{^}

and

\overset{v}{^}

unit vectors this reduces to

6 sin θ = 1

, that is

sin θ = \frac{1}{6}

. Restricted to the acute range, exactly one angle

θ

satisfies this.

JEE Main · 2007 · 29 Apr 2007 · Offline · Single correct option