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Standard XII

Mathematics

Contra-Positive

If a, b, c ar...

Question

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

l + m + n

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

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

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None of these

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Solution

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, $p . a = 1$ , $p . b = 0 = p . c$
$q . a = 0, q . b = 1, q . c = 0$
$r . a = 0$ , $r . b = 0$ , $r . c = 1$
$∴ (l a + m b + n c) . (l p + m q + n r)$
$= l^{2} a \cdot p + m^{2} b \cdot q + n^{2} c \cdot r$
$= l^{2} + m^{2} + n^{2}$
Hence, option C is correct.

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