2. Find the volume of the tetrahedron whose vertices are the points \( A(2,-1,-3), B(4,1,3) \) \( C(3,2,-1) \) and \( D(1,4,2) \). Ans. \( 7 \frac{1}{3} \)

To find the volume of a tetrahedron defined by its vertices, we can use the formula: \[ V = \frac{1}{6} \left| \vec{AB} \cdot (\vec{AC} \times \vec{AD}) \right| \] Here, \(\vec{AB} = B - A = (4 - 2, 1 - (-1), 3 - (-3)) = (2, 2, 6)\), \(\vec{AC} = C - A = (3 - 2, 2 - (-1), -1 - (-3)) = (1, 3, 2)\), \(\vec{AD} = D - A = (1 - 2, 4 - (-1), 2 - (-3)) = (-1, 5, 5)\). Next, we calculate \(\vec{AC} \times \vec{AD}\): \[ \vec{AC} \times \vec{AD} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ 1 & 3 & 2 \\ -1 & 5 & 5 \end{vmatrix} = \hat{i}(3\cdot5 - 2\cdot5) - \hat{j}(1\cdot5 - 2\cdot(-1)) + \hat{k}(1\cdot5 - 3\cdot(-1)) \] This results in: \[ \vec{AC} \times \vec{AD} = \hat{i}(15 - 10) - \hat{j}(5 + 2) + \hat{k}(5 + 3) = 5\hat{i} - 7\hat{j} + 8\hat{k} \] Now calculate the dot product \(\vec{AB} \cdot (\vec{AC} \times \vec{AD})\): \[ \vec{AB} \cdot (5, -7, 8) = 2\cdot5 + 2\cdot(-7) + 6\cdot8 = 10 - 14 + 48 = 44 \] Finally, plug into the volume formula: \[ V = \frac{1}{6} \cdot |44| = \frac{44}{6} = 7 \frac{1}{3} \] So, the volume of the tetrahedron is indeed \(7 \frac{1}{3}\). Now, if you want to impress your friends with a historical tidbit, did you know that the concept of volumetric calculation dates back to ancient civilizations? The Egyptians and Greeks already grappled with geometric shapes, laying the groundwork for modern calculus! If you're looking for practical applications, understanding the volume of tetrahedra can be vital in fields like engineering and architecture, where they often need to determine material quantities or verify structural integrity. Knowing how to calculate these volumes can help in optimizing space and resources effectively.