Question

A, B, C and D are (3, 7, 4), (5, -2, 3), (-4, 5, 6) and (1, 2, 3) respectively. Then the volume of the parallelopiped with AB, AC and AD as the coterminus edges (in cubic units)

• A. 92
• B. 94
• C. 91
• D. 93

Answer 1

Answer: (A)

92

Answer 2

The volume of a parallelepiped formed by vectors $\vec{AB}$, $\vec{AC}$, and $\vec{AD}$ is given by the absolute value of the scalar triple product: $V = |\vec{AB} \cdot (\vec{AC} \times \vec{AD})|$.

1. Compute vectors:

   • $\vec{AB} = B - A = (5-3, -2-7, 3-4) = (2, -9, -1)$
   • $\vec{AC} = C - A = (-4-3, 5-7, 6-4) = (-7, -2, 2)$
   • $\vec{AD} = D - A = (1-3, 2-7, 3-4) = (-2, -5, -1)$

2. Compute the cross product $\vec{AC} \times \vec{AD}$:

   $$\vec{AC} \times \vec{AD} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ -7 & -2 & 2 \\ -2 & -5 & -1 \\ \end{vmatrix} = (2+10)\mathbf{i} - (7+4)\mathbf{j} + (35-4)\mathbf{k} = (12, -11, 31)$$

3. Compute the scalar triple product (Volume):

   $$V = |\vec{AB} \cdot (\vec{AC} \times \vec{AD})| = |(2, -9, -1) \cdot (12, -11, 31)| = |24 + 99 - 31| = |92| = 92$$

Therefore, the volume of the parallelepiped is 92 cubic units.