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Applications Scalar Triple Product

If V is the...

Question

If $V$ is the volume of parallelepiped formed by the vectors $\to a, \to b, \to c$ as three coterminous edges is $27$ cubic units, then the volume of the parallelepiped having $\to α = \to a + \to 2 b - \to c, \to β = \to a - \to b$ and $\to γ = \to a - \to b - \to c$ as three coterminous edges is

A

27

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B

9

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C

81

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D

48

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Solution

The correct option is A $27$
Given that
$[\to a \to b \to c] = 27 = V$ (volume)

$\to α = \to a + 2 \to b - \to c$
$\to β = \to a - \to b + 0 \to c$
$\to γ = \to a - \to b - \to c$

$[\to α \to β \to γ] =$ $∣ ∣ ∣ ∣ \begin{matrix} 1 & 2 & - 1 1 & - 1 & 0 1 & - 1 & - 1 \end{matrix} ∣ ∣ ∣ ∣$ $[\to a \to b \to c]$

$= 3 [\to a \to b \to c]$
$= 27 \times 3 = 81$ cubic units

Hence, option C is the correct option.

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