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Vector Addition

If V is the...

Question

If $V$ is the volume of parallelepiped formed by the vectors $\to a, \to b, \to c$ as three coterminuous edges is $27$ cubic units, then the volume of the parallelrpiped having $\to a = \to a + 2 \to b - \to c, \to β = \to a - \to b$ 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 C $81$
We have given from the question
$\frac{1}{6} [\to a \to b \to c] = 27$

$V = \frac{1}{6} [\to a + 2 \to b - \to c \to a - \to b \to a - \to b - \to c]$
$= \frac{1}{6} ∣ ∣ ∣ ∣ \begin{matrix} 2 & - 1 1 & - 1 & 0 1 & - 1 & - 1 \end{matrix} ∣ ∣ ∣ ∣ [\to a \to b \to c]$
$= 27 [1 (1 - 0) - 2 (- 1) - 1 (- 1 + 1)]$
$= 27 [1 + 2]$
$= 27 \times 3$
$= 81$

Hence, the option $(C)$ is the correct answer.

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