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Vector Triple Product

If a⃗ = i⃗ ...

Question

If $\to a = \to i + 2 \to j + \to k, \to b = 2 i - j + k, \to c = j + 2 k$ and if $\to s$ is any unit vector in space then the value of magnitude of $(\to s . \to a) (\to b \times \to c) + (\to s . \to b) (\to c \times \to a) + (\to s . \to c) (\to a \times \to b)$ equals

A

9

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B

\frac{9}{4}

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C

4

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D

1

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Solution

The correct option is A $9$

$\to b \times \to c = - 3^ı - 4^j + 2^k$
$\to c \times \to a = - 3^i + 2^ȷ -^k$
$\to a \times \to h = 2^ı +^ı$
$\to a \times \to b = 3^i + {^ȷ}_{1}^{2}^k$
Let where
$\to s \cdot \to a = p + 2 q + r$
$\to s \cdot \to b = 2 p - q + μ$
$\to s \cdot \to c = q + 2 k$
$\Rightarrow (\to s \cdot \to a) (\to b \times \to c) + (\to s \cdot \to b) (\to c \times \to a) + (\to s \cdot \to c) (a \times \to b)$
$= (p + 2 q + r) (- 3^i - 4^ȷ + 2^k) + (2 p - q + u) x$
$\begin{matrix} (- 3^i + 2^j -^k) + (q + 2 x) (3^ı +^ȷ - 5^k) = & (- 9 p)^ı + (- 9 q)^j + (- 9 n)^k \end{matrix}$
$= 9$
$∴$ option $A$ is correct

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