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Scalar Triple Product

If a,b,c are ...

Question

If $\to a, \to b, \to c$ are three non coplanar vectors and a vector $\to α$ is such that $\to α = p (\to b \times \to c) + q (\to c \times \to a) + r (\to a \times \to b) and \to α \cdot (\to a + \to b + \to c) = 1$ $, then [\to a \to b \to c]$ is equal to

A

p + q + r

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B

\frac{1}{p + q + r}

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C

2 (p + q + r)

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D

\frac{2}{p + q + r}

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Solution

The correct option is B $\frac{1}{p + q + r}$
Given: $\to α = p (\to b \times \to c) + q (\to c \times \to a) + r (\to a \times \to b)$ and
$\to α \cdot (\to a + \to b + \to c) = 1$
Now,
$\to α \cdot \to a = p [\to a \to b \to c]$
$\to α \cdot \to b = q [\to a \to b \to c]$
$\to α \cdot \to c = r [\to a \to b \to c]$
Adding them, we have:
$\to α \cdot (\to a + \to b + \to c) = (p + q + r) [\to a \to b \to c]$
$\Rightarrow 1 = (p + q + r) [\to a \to b \to c]$ $\Rightarrow [\to a \to b \to c] = \frac{1}{p + q + r}$

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Scalar Triple Product

Standard XII Mathematics

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