For non-zero vectors a, b and c the relation |(a x b).c| = |a||b||c| holds good, if A. a.b = b.c =0 B. a.b = 0 = c.a C. a.b = b.c = c.a = 0

Question

For non-zero vectors a, b and c the relation |(a x b).c| = |a||b||c| holds good, if 

A. \(\vec a.\vec b=\vec b.\vec c=0\)

B. \(\vec a.\vec b = 0 = \vec c.\vec a\)

C. \(\vec a.\vec b=\vec b.\vec c=\vec c.\vec a=0\)

D. \(\vec b.\vec c=\vec c.\vec a=0\)

Answer 1

Let  \(\vec e=\vec a\times\vec b\)

\(|\vec e|=|\vec a||\vec b|sin \alpha\)....(1) 

(∵ α is angle between  \(\vec a\) and  \(\vec b\))

Then 

\(=|\vec e||\vec c|cos\theta\)

(∵θ is angle between  \(\vec e\)and  \(\vec c\) ⇒ θ is angle between \(\vec a\times\vec b\) and  \(\vec c\))

\(\alpha=\cfrac{\pi}2\implies\vec a\) and  \(\vec b\) are perpendicular.

Also  \(\vec e\) is perpendicular to both  \(\vec a\) and  \(\vec b\).

θ = 0 ⇒  \(\vec c\)is perpendicular to both  \(\vec a\) and  \(\vec b\).

\(\therefore\vec a,\,\vec b,\,\vec c\) are mutually perpendicular.

\(\therefore\vec a.\vec b=\vec b.\vec c=\vec c.\vec a=0\)