JEE MAIN - Mathematics (2007 - No. 16)
If $$\widehat u$$ and $$\widehat v$$ are unit vectors and $$\theta $$ is the acute angle between them, then $$2\widehat u \times 3\widehat v$$ is a unit vector for :
no value of $$\theta $$
exactly one value of $$\theta $$
exactly two values of $$\theta $$
more than two values of $$\theta $$
Explanation
Given $$\left| {2\widehat u \times 3\widehat v} \right| = 1$$
and $$\theta $$ is acute angle between $$\widehat u$$
and $$\widehat v,\,\,\left| {\widehat u} \right| = 1,\,\,\left| {\widehat v} \right| = 1\,\,\,$$
$$ \Rightarrow \,\,\,6\left| {\widehat u} \right|\left| {\widehat v} \right|\left| {\sin \theta } \right| = 1$$
$$ \Rightarrow 6\left| {\sin \theta } \right| = 1 \Rightarrow \sin \theta = {1 \over 6}$$
Hence, there is exactly one value of $$\theta $$
for which $$2\widehat u \times 3\widehat v$$ is a unit vector.
and $$\theta $$ is acute angle between $$\widehat u$$
and $$\widehat v,\,\,\left| {\widehat u} \right| = 1,\,\,\left| {\widehat v} \right| = 1\,\,\,$$
$$ \Rightarrow \,\,\,6\left| {\widehat u} \right|\left| {\widehat v} \right|\left| {\sin \theta } \right| = 1$$
$$ \Rightarrow 6\left| {\sin \theta } \right| = 1 \Rightarrow \sin \theta = {1 \over 6}$$
Hence, there is exactly one value of $$\theta $$
for which $$2\widehat u \times 3\widehat v$$ is a unit vector.
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