Question

If $A,b,c$ are three non zero vectors then show that
$|(a\times b).c|=\left|a\right|\left|b\right|\left|c\right|ifa.b=b.c=c.a=0$

Answer 1

$a.b=b.c=c.a=0$

$=a{\perp }_{r}b$

$=b{\perp }_{r}c$

$c{\perp }_{r}a$

$|(a\times b).c|=|a\times b||c|\cos (a\times b^{n}c)$

$a\times bis{\perp }_r$ to a and b

but C is also ${\perp }_r$ to a and b

$\therefore$ c is parallel to $\overline{a}\times \overline{b}$

$\cos (a\times b^{n}c)=1$

$=|a\times b||c|$

$|a||b||c|\sin {90}^o$

$|a||b||c|$