Question
For non-zero vectors $$\vec a,\,\vec b,\,\vec c,\,\left| {\left( {\vec a \times \vec b} \right).\vec c} \right| = \left| {\vec a} \right|\left| {\vec b} \right|\left| {\vec c} \right|$$ holds if and only if -
A.
$$\vec a.\vec b = 0,\,\,\,\vec b.\vec c = 0$$
B.
$$\vec b.\vec c = 0,\,\,\,\vec c.\vec a = 0$$
C.
$$\vec c.\vec a = 0,\,\,\,\vec a.\vec b = 0$$
D.
$$\vec a.\vec b = \vec b.\vec c = \vec c.\vec a = 0$$
Answer :
$$\vec a.\vec b = \vec b.\vec c = \vec c.\vec a = 0$$
Solution :
$$\eqalign{
& \left| {\left( {\vec a \times \vec b} \right).\vec c} \right| = \left| {\vec a} \right|\left| {\vec b} \right|\left| {\vec c} \right| \cr
& \Rightarrow \left| {\left| {\hat a} \right|\left| {\hat b} \right|\sin \,\theta \,\hat n.\,\vec c} \right|\, = \left| {\hat a} \right|\left| {\hat b} \right|\left| {\hat c} \right| \cr
& {\text{where }}\theta {\text{ is angle between }}\vec a{\text{ and }}\vec b \cr
& \Rightarrow \left| {\hat a} \right|\left| {\hat b} \right|\left| {\hat c} \right|\,\left| {\sin \,\theta \,\cos \,\alpha } \right| = \left| {\hat a} \right|\left| {\hat b} \right|\left| {\hat c} \right| \cr
& {\text{where }}\alpha \,{\text{is angle betwen }}\vec c{\text{ and }}\hat n \cr
& \Rightarrow \left| {\sin \,\theta \,} \right|\,\left| {\cos \,\alpha } \right| = 1 \cr
& \Rightarrow \theta = \frac{\pi }{2}{\text{ and }}\alpha = 0 \cr
& \Rightarrow \vec a \bot \vec b{\text{ and }}\vec c\left\| {\hat n} \right. \cr
& \Rightarrow \vec a.\vec b = \vec b.\vec c = \vec c.\vec a = 0 \cr} $$