Question
For three noncoplanar vectors $$\overrightarrow a ,\,\overrightarrow b ,\,\overrightarrow c $$ the relation $$\left| {\overrightarrow a \times \overrightarrow b .\overrightarrow c } \right| = \left| {\overrightarrow a } \right|\left| {\overrightarrow b } \right|\left| {\overrightarrow c } \right|$$ holds if and only if :
A.
$$\overrightarrow b .\overrightarrow c = \overrightarrow c .\overrightarrow a = 0$$
B.
$$\overrightarrow a .\overrightarrow b = \overrightarrow b .\overrightarrow c = 0$$
C.
$$\overrightarrow a .\overrightarrow b = \overrightarrow b .\overrightarrow c = \overrightarrow c .\overrightarrow a = 0$$
D.
$$\overrightarrow c .\overrightarrow a = \overrightarrow a .\overrightarrow b = 0$$
Answer :
$$\overrightarrow a .\overrightarrow b = \overrightarrow b .\overrightarrow c = \overrightarrow c .\overrightarrow a = 0$$
Solution :
$$\left| {\overrightarrow a \times \overrightarrow b .\overrightarrow c } \right| = $$ volume of the parallelepiped $$ = \left| {\overrightarrow a } \right|\left| {\overrightarrow b } \right|\left| {\overrightarrow c } \right|$$
$$ \Rightarrow $$ it is a rectangular parallelepiped, i.e., concurrent edges are perpendicular to each other. So, $$\overrightarrow a .\overrightarrow b = 0,\,\overrightarrow b .\overrightarrow c = 0,\,\overrightarrow c .\overrightarrow a = 0.$$