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Dot Product of Two Vectors

If a= 2p̂+3...

Question

If $a = 2^p + 3^q -^r, b =^p - 2^q + 2^r$ and $c = - 2^p +^q - 2^r$ and $R = 3^p -^q + 2^r,$ express $R$ in terms of $a, b, c,$ when $^p,^q,^r$ are non-coplanar vectors.

R = a + 2 b + 3 c

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R = 2 a + 3 b + c

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R = 3 a + 5 b + 2 c

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R = 2 a + 5 b + 3 c

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Solution

The correct option is C $R = 2 a + 5 b + 3 c$
Let us suppose that $R = x a + y b + z c$ ...(1)
$\Rightarrow R = x (2, 3, - 1) + y (1, - 2, 2) + z (- 2, 1, - 2)$
$= (2 x + y - 2 z) p + (3 x - 2 y + z) q + (- x + 2 y - 2 z) r$
$= 3 p - q + 2 r$ (given)
Comparing, we get $2 x + y - 2 z = 3, 3 x - 2 y + z = - 1, - x + 2 y - 2 z = 2$
Solving these equations we get, $x = 2, y = 5, z = 3$
$∴ R = 2 a + 5 b + 3 c$

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