Question

If a, b, c are three non-coplanar vectors and p, q, r are reciprocal vectors, then $(la+mb+nc).(lp+mq+nr)$ is equal to

• A. $l+m+n$
• B. $l^3+m^3+n^3$
• C. $l^2+m^2+n^2$
• D. None of these

Answer 1

The correct option is C $l^2+m^2+n^2$

p, q and r are reciprocal vectors of a, b and c respectively.
So, $p.a=1$, $p.b=0=p.c$
$q.a=0,q.b=1,q.c=0$
$r.a=0$, $r.b=0$, $r.c=1$
$\therefore (la+mb+nc).(lp+mq+nr)$

$=l^{2}a\cdot p+m^{2}b\cdot q+n^{2}c\cdot r$
$=l^2+m^2+n^2$

Hence, option C is correct.