If a→,b→,c→ are three $$non-coplanar$$ and p→,q→,r→ are reciprocal vectors to a→,b→ and c→ respectively, then la→$$+mb$$→$$+nc$$→⋅lp→$$+mq$$→$$+nr$$→ is equal to : (where$$ $$1$$,m,$$n_$$ are $$scalars)

Question

If a→,b→,c→ are three $$non-coplanar$$ and p→,q→,r→ are reciprocal vectors to a→,b→ and c→ respectively, then la→$$+mb$$→$$+nc$$→⋅lp→$$+mq$$→$$+nr$$→ is equal to : (where$$ $$1$$,m,$$n_$$ are $$scalars)

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p→   $$=$$  b→ × c→a→   b→   c→  ;  q→  $$=$$  c→ × a→a→   b→   c→ ;  r→  $$=$$ a→ × b→a→   b→   c→a→.p→$$=a$$→·  b→ × c→a→   b→   c→$$=a$$→   b→   c→a→   b→   c→$$=1$$    and a→.q→$$=a$$→·  c→ × a→a→   b→   c→$$=a$$→   c→   a→a→   b→   c→$$=0similarly$$ b→·q→ $$=1$$  c→·r→$$=1$$ la→$$+mb$$→$$+nc$$→⋅lp→$$+mq$$→$$+nr$$→ $$=l2a$$→·p→ $$+m2b$$→.q→$$+n2c$$→·r→  $$=l2+m2+n2$$

$If \vec{a}, \vec{b}, \vec{c} are three non-coplanar and \vec{p}, \vec{q}, \vec{r} are reciprocal vectors to \vec{a}, \vec{b}$ $and \vec{c} respectively, then (l \vec{a} + m \vec{b} + n \vec{c}) \cdot (l \vec{p} + m \vec{q} + n \vec{r}) is equal to : (where$ $1, m, \underline{n} are scalars)$

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If a→,b→,c→ are three non-coplanar and p→,q→,r→ are reciprocal vectors to a→,b→ and c→ respectively, then la→+mb→+nc→⋅lp→+mq→+nr→ is equal to : (where 1,m,n_ are scalars)

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$If \vec{a}, \vec{b}, \vec{c} are three non-coplanar and \vec{p}, \vec{q}, \vec{r} are reciprocal vectors to \vec{a}, \vec{b}$ $and \vec{c} respectively, then (l \vec{a} + m \vec{b} + n \vec{c}) \cdot (l \vec{p} + m \vec{q} + n \vec{r}) is equal to : (where$ $1, m, \underline{n} are scalars)$