What is the volume of the parallelepiped defined by the vectors \( \left[\begin{array}{l}8 \\ 5 \\ 0\end{array}\right],\left[\begin{array}{c}0 \\ 6 \\ -3\end{array}\right],\left[\begin{array}{c}-4 \\ -7 \\ 9\end{array}\right] \) ? Volume \( = \) Ex. 1 What is the volume of the tetrahedron defined by the vectors \( \left[\begin{array}{c}8 \\ 5 \\ 0\end{array}\right],\left[\begin{array}{c}0 \\ 6 \\ -3\end{array}\right],\left[\begin{array}{c}-4 \\ -7 \\ 9\end{array}\right] \) ?

Let the vectors be \[ \mathbf{v}_1 = \begin{bmatrix} 8 \\ 5 \\ 0 \end{bmatrix}, \quad \mathbf{v}_2 = \begin{bmatrix} 0 \\ 6 \\ -3 \end{bmatrix}, \quad \mathbf{v}_3 = \begin{bmatrix} -4 \\ -7 \\ 9 \end{bmatrix}. \] ### Step 1. Compute the Scalar Triple Product The volume of the parallelepiped is given by the absolute value of the scalar triple product: \[ V_{\text{parallelepiped}} = \left| \mathbf{v}_1 \cdot \left( \mathbf{v}_2 \times \mathbf{v}_3 \right) \right|. \] #### a. Compute the Cross Product \(\mathbf{v}_2 \times \mathbf{v}_3\) Using the formula \[ \mathbf{a} \times \mathbf{b} = \begin{bmatrix} a_y b_z - a_z b_y \\ a_z b_x - a_x b_z \\ a_x b_y - a_y b_x \end{bmatrix}, \] we let \(\mathbf{a} = \mathbf{v}_2\) and \(\mathbf{b} = \mathbf{v}_3\). Thus, \[ \mathbf{v}_2 \times \mathbf{v}_3 = \begin{bmatrix} 6 \cdot 9 - (-3) \cdot (-7) \\ (-3) \cdot (-4) - 0 \cdot 9 \\ 0 \cdot (-7) - 6 \cdot (-4) \end{bmatrix}. \] Calculating each component: - **First component:** \[ 6\cdot9 - (-3)\cdot(-7) = 54 - 21 = 33. \] - **Second component:** \[ (-3)\cdot(-4) - 0\cdot9 = 12 - 0 = 12. \] - **Third component:** \[ 0\cdot(-7) - 6\cdot(-4) = 0 + 24 = 24. \] Thus, \[ \mathbf{v}_2 \times \mathbf{v}_3 = \begin{bmatrix} 33 \\ 12 \\ 24 \end{bmatrix}. \] #### b. Compute the Dot Product \(\mathbf{v}_1 \cdot (\mathbf{v}_2 \times \mathbf{v}_3)\) \[ \mathbf{v}_1 \cdot \begin{bmatrix} 33 \\ 12 \\ 24 \end{bmatrix} = 8 \cdot 33 + 5 \cdot 12 + 0 \cdot 24. \] Calculating, \[ 8 \cdot 33 = 264, \quad 5 \cdot 12 = 60, \quad 0 \cdot 24 = 0. \] Thus, \[ \mathbf{v}_1 \cdot (\mathbf{v}_2 \times \mathbf{v}_3) = 264 + 60 + 0 = 324. \] ### Step 2. Determine the Volume of the Parallelepiped Taking the absolute value, \[ V_{\text{parallelepiped}} = \left| 324 \right| = 324. \] ### Step 3. Determine the Volume of the Tetrahedron The volume of the tetrahedron defined by the same three vectors is one-sixth of the parallelepiped volume: \[ V_{\text{tetrahedron}} = \frac{1}{6} V_{\text{parallelepiped}} = \frac{324}{6} = 54. \] ### Final Answer - The volume of the parallelepiped is \(324\). - The volume of the tetrahedron is \(54\).