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Slope Intercept Form of a Line

If A 1,2 , ...

Question

If $A (1, 2, - 1) B (3, 5, 0) C (5, 11, 22)$ and $D (5, 8, 6)$ are the vertices. Find the volume of the parallelepiped with segment $A B, A C$ and $A D$ as concurrent edges.

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Solution

Volume of parallelopiped is $∣ ∣ ∣ - - \to A B . (- - \to A C \times - - \to A D) ∣ ∣ ∣$
now
$- - \to A B = (3 - 1) ˆ i + (5 - 2) ˆ j + (0 - (- 1)) ˆ k = 2 ˆ i + 3 ˆ j + 1 ˆ k - - \to A C = (5 - 1) ˆ i + (11 - 2) ˆ j + (22 - (- 1)) ˆ k = 4 ˆ i + 9 ˆ j + 23 ˆ k - - \to A D = (5 - 1) ˆ i + (8 - 2) ˆ j + (6 - (- 1)) ˆ k = 4 ˆ i + 6 ˆ j + 7 ˆ k$
so volume of the parallelopiped is
$∣ ∣ (2 ˆ i + 3 ˆ j + ˆ k) . ((4 ˆ i + 9 ˆ j + 23 ˆ k) \times (4 ˆ i + 6 ˆ j + 7 ˆ k)) ∣ ∣ = ∣ ∣ (2 ˆ i + 3 ˆ j + ˆ k) . ((63 - 138) ˆ i + (- 28 + 92) ˆ j + (24 - 36) ˆ k) ∣ ∣ = ∣ ∣ (2 ˆ i + 3 ˆ j + ˆ k) . (- 75 ˆ i + ˆ 64 j - 12 ˆ k) ∣ ∣ = | - 150 + 192 - 12 | = 30$

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