Question
$$\left[ {\overrightarrow b \,\,\overrightarrow c \,\,\overrightarrow b \times \overrightarrow c } \right] + {\left( {\overrightarrow b .\overrightarrow c } \right)^2}$$ is equal to :
A.
$${\left| {\overrightarrow b } \right|^2}{\left| {\overrightarrow c } \right|^2}$$
B.
$${\left( {\overrightarrow b + \overrightarrow c } \right)^2}$$
C.
$${\left| {\overrightarrow b } \right|^2} + {\left| {\overrightarrow c } \right|^2}$$
D.
none of these
Answer :
$${\left| {\overrightarrow b } \right|^2}{\left| {\overrightarrow c } \right|^2}$$
Solution :
$$\eqalign{
& \left[ {\overrightarrow b \,\,\overrightarrow c \,\,\overrightarrow b \times \overrightarrow c } \right] = \overrightarrow b .\overrightarrow c \times \left( {\overrightarrow b \times \overrightarrow c } \right) \cr
& = \overrightarrow b .\left\{ {\left( {\overrightarrow c .\overrightarrow c } \right)\overrightarrow b - \left( {\overrightarrow c .\overrightarrow b } \right)\overrightarrow c } \right\} \cr
& = \left( {\overrightarrow c .\overrightarrow c } \right)\left( {\overrightarrow b .\overrightarrow b } \right) - {\left( {\overrightarrow b + \overrightarrow c } \right)^2} \cr
& \therefore \,\left[ {\overrightarrow b \,\,\overrightarrow c \,\,\overrightarrow b \times \overrightarrow c } \right] + {\left( {\overrightarrow b + \overrightarrow c } \right)^2} = {\left| {\overrightarrow c } \right|^2}\,{\left| {\overrightarrow b } \right|^2} \cr} $$