Question
Let $$\overrightarrow a $$ be a unit vector perpendicular to unit vectors $$\overrightarrow b $$ and $$\overrightarrow c $$ and if the angle between $$\overrightarrow b $$ and $$\overrightarrow c $$ be $$\alpha $$ then $$\overrightarrow b \times \overrightarrow c $$ is :
A.
$$\cos \,\alpha \,\overrightarrow a $$
B.
$${\text{cosec}}\,\alpha \,\overrightarrow a $$
C.
$$\sin \,\alpha \,\overrightarrow a $$
D.
none of these
Answer :
$$\sin \,\alpha \,\overrightarrow a $$
Solution :
$$\eqalign{
& \left| {\overrightarrow b \times \overrightarrow c } \right| = \left| {\overrightarrow b } \right|\left| {\overrightarrow c } \right|\,\sin \,\alpha = \sin \,\alpha \cr
& {\text{Now, }}\frac{{\overrightarrow b \times \overrightarrow c }}{{\left| {\overrightarrow b \times \overrightarrow c } \right|}} = \overrightarrow a \,\,\left( {{\text{given}}} \right) \cr
& \therefore \,\,\overrightarrow b \times \overrightarrow c = \sin \,\alpha \,\overrightarrow a . \cr} $$