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Scalar Triple Product

If a, b, c ar...

Question

If $\to a, \to b, \to c$ are three non coplanar vectors and a vector $\to r = \frac{(\to r . \to a) (\to b \times \to c) + (\to r . \to b) (\to c \times \to a) + (\to r . \to c) (\to a \times \to b)}{λ},$ then the value of $λ$ is:

A

[\to a \to c \to b]

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B

\frac{1}{[\to a \to b \to c]}

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C

[\to c \to a \to b]

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D

[2 \to b \to a \to c]

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Solution

The correct option is C $[\to c \to a \to b]$
As $\to a, \to b, \to c$ are three non coplanar vectors, then $(\to a \times \to b), (\to b \times \to c), (\to c \times \to a)$ are also non coplanar.
Any vector $\to r$ can be written in terms of three non coplanar vectors as:
$\to r = l (\to a \times \to b) + m (\to b \times \to c) + n (\to c \times \to a)$ where $l, m, n$ are scalar.
Now, $\to r . \to a = m [\to a \to b \to c]$
$\Rightarrow m = \frac{\to r . \to a}{[\to a \to b \to c]}$
Similarly, $n = \frac{\to r . \to b}{[\to a \to b \to c]}$
and $l = \frac{\to r . \to c}{[\to a \to b \to c]}$ Hence on comparing, we have $λ = [\to a \to b \to c]$

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Scalar Triple Product

Standard XII Mathematics

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