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

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) l3 + m3 + n3 (C) l2 + m2 + n2  (D) None of these

Tardigrade Admin · Accepted Answer

∵ p, q and r are reciprocal vectors of a, b and c respectively. ∴ p ⋅ r =1, p ⋅ b = 0 = p ⋅ c q ⋅ a =0, q ⋅ b =1, q ⋅ c =0 r ⋅ a =0, r ⋅ b =0, r ⋅ c =1 ∴ (l a +m b +n c ) ⋅(l p +m q +n r ) =l2+m2+n2

Q. If $a, b, c$ are three non-coplanar vectors and $p, q, r$ are reciprocal vectors, then $(l a + mb + n c) . (lp + m q + n r)$ is equal to

$l + m + n$

$l^{3} + m^{3} + n^{3}$

$l^{2} + m^{2} + n^{2}$

None of these

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

Q. 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

l+m+n

l3+m3+n3

l2+m2+n2