Find the volume of a parallelopiped whose edges are represented by the vectors :
$\to a = 2^i - 3^j - 4^k, \to b =^i + 2^j -^k,$ and $\to c = 3^i +^j + 2^k$ .

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Solution

Given,
$\to a = 2^i - 3^j - 4^k$
$\to b =^i + 2^j -^k$
$\to c = 3^i +^j + 2^k$

$[\to a \to b \to c] = ∣ ∣ ∣ ∣ \begin{matrix} 2 & - 3 & - 4 1 & 2 & - 1 3 & 1 & 2 \end{matrix} ∣ ∣ ∣ ∣$

Expanding along first row, we get
$\Rightarrow 2 ∣ ∣ ∣ \begin{matrix} 2 & - 1 1 & 2 \end{matrix} ∣ ∣ ∣ - (- 3) ∣ ∣ ∣ \begin{matrix} 1 & - 1 3 & 2 \end{matrix} ∣ ∣ ∣ + (- 4) ∣ ∣ ∣ \begin{matrix} 1 & 2 3 & 1 \end{matrix} ∣ ∣ ∣$
$= 2 (4 + 1) + 3 (2 + 3) - 4 (1 - 6)$
$= 2 \times 5 + 3 \times 5 - 4 (- 5)$

Ans. $= 10 + 15 + 20 = 45$ .

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