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Vector Triple Product

Let a,b,c be ...

Question

Let $\to a, \to b, \to c$ be three vectors such that $| \to a | = | \to b | = | \to c | = 4$ and angle between $\to a$ and $\to b$ is $π / 3$ angle between $\to b$ and $\to c$ is $π / 3$ and angle between $\to c$ and $\to a$ is $π / 3.$
The volume of trianglular prism whose adjacent edges are represented by the vectors $\to a, \to b$ and $\to c$

A

12 \sqrt{2}

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B

12 \sqrt{3}

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C

16 \sqrt{2}

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D

16 \sqrt{3}

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Solution

The correct option is C $16 \sqrt{2}$
Volume of the parallelopiped $= [\to a \to b \to c]$
Let $[\to a \to b \to c] = k$
$[\to a \to b \to c]^{2} = ∣ ∣ ∣ ∣ \begin{matrix} a_{1} & a_{2} & a_{3} b_{1} & b_{2} & b_{3} c_{1} & c_{2} & c_{3} \end{matrix} ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ \begin{matrix} a_{1} & a_{2} & a_{3} b_{1} & b_{2} & b_{3} c_{1} & c_{2} & c_{3} \end{matrix} ∣ ∣ ∣ ∣$
$[\to a \to b \to c]^{2} = ∣ ∣ ∣ ∣ ∣ ∣ \begin{matrix} \to a . \to a & \to a . \to b & \to a . \to c \to b . \to a & \to b . \to b & \to b . \to c \to c . \to a & \to c . \to b & \to c . \to c \end{matrix} ∣ ∣ ∣ ∣ ∣ ∣$
$k^{2} = ∣ ∣ ∣ ∣ ∣ ∣ \begin{matrix} | \to a |^{2} & | \to a | | \to b | cos π / 3 & | \to a | | \to c | cos π / 3 | \to b | | \to a | cos π / 3 & | \to b |^{2} & | \to b | | \to c | cos π / 3 | \to c | | \to a | cos π / 3 & | \to c | | \to b | cos π / 3 & | \to c |^{2} \end{matrix} ∣ ∣ ∣ ∣ ∣ ∣$
$k^{2} = ∣ ∣ ∣ ∣ \begin{matrix} 16 & 8 & 8 8 & 16 & 8 8 & 8 & 16 \end{matrix} ∣ ∣ ∣ ∣$
$k^{2} = 8^{3} ∣ ∣ ∣ ∣ \begin{matrix} 2 & 1 & 1 1 & 2 & 1 1 & 1 & 2 \end{matrix} ∣ ∣ ∣ ∣ = 8^{3} \times 4$
$k = 32 \sqrt{2}$
Hence, volume of parallelopiped $= [\to a \to b \to c] = 32 \sqrt{2}$
Voulume of triangular prism $= \frac{1}{2} [\to a \to b \to c] = \frac{1}{2} \times 32 \sqrt{2}$
Voulume of triangular prism $= 16 \sqrt{2}$

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Vector Triple Product

Standard XII Mathematics

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