Find the volume of the parallelopiped whose coterminus edges are given by vectors $2^i + 3^j - 4^k, 5^j + 7^j + 5^k$ and $4^i + 4^j - 2^k$

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Solution

Let coterminus edges are $\to a = 2^i + 3^j - 4^k, \to b = 5^i + 7^j + 5^k$ and $\to c = 4^i + 4^j - 2^k$

So by definition volume of parallelopiped $= | [\to a \to b \to c] | = ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ \begin{matrix} 2 & 3 & - 4 5 & 7 & 5 4 & 4 & - 2 \end{matrix} ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣$

$= | 2 (- 14 - 20) - 3 (- 10 - 20) - 4 (20 - 28) |$ , expand the determinant along first row

$= | - 68 + 90 + 32 | = 54$

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