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

Answer: (C)

$l^2 + m^2 + n^2$

Answer 2

$\because p, q$ and $r$ are reciprocal vectors of $a, b$ and $c$ respectively.
$\therefore p \cdot r =1, p \cdot b = 0 = p \cdot c$
$q \cdot a =0, q \cdot b =1, q \cdot c =0$
$r \cdot a =0, r \cdot b =0, r \cdot c =1$
$\therefore (l a +m b +n c ) \cdot(l p +m q +n r )$
$=l^{2}+m^{2}+n^{2}$