Question
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.
no value of $$\theta $$
B.
exactly one value of $$\theta $$
C.
exactly two values of $$\theta $$
D.
more than two values of $$\theta $$
Answer :
exactly one value of $$\theta $$
Solution :
Given $$\left| {2\hat u \times 3\hat v} \right| = 1$$ and $$\theta $$ is acute angle between $${\hat u}$$ and $$\hat v,\,\,\left| {\hat u} \right| = 1,\,\,\left| {\hat v} \right| = 1$$
$$\eqalign{
& \Rightarrow 6\,\left| {\hat u} \right|\,\left| {\hat v} \right|\left| {\sin \,\theta } \right| = 1 \cr
& \Rightarrow 6\left| {\sin \,\theta } \right| = 1 \cr
& \Rightarrow \sin \,\theta = \frac{1}{6} \cr} $$
Hence, there is exactly one value of $$\theta $$ for which $$2\hat u \times 3\hat v$$ is a unit vector.