What is the volume of the parallelepiped defined by the vectors \( \left[\begin{array}{c}-2 \\ 4 \\ 8\end{array}\right],\left[\begin{array}{c}-8 \\ 5 \\ 0\end{array}\right],\left[\begin{array}{l}0 \\ 2 \\ 9\end{array}\right] \) Volume \( = \) Ex. 1

1. The volume \( V \) of the parallelepiped defined by vectors \( \mathbf{a} \), \( \mathbf{b} \), and \( \mathbf{c} \) is given by: \[ V = \left| \mathbf{a} \cdot (\mathbf{b} \times \mathbf{c}) \right| \] 2. Given the vectors: \[ \mathbf{a} = \begin{pmatrix} -2 \\ 4 \\ 8 \end{pmatrix}, \quad \mathbf{b} = \begin{pmatrix} -8 \\ 5 \\ 0 \end{pmatrix}, \quad \mathbf{c} = \begin{pmatrix} 0 \\ 2 \\ 9 \end{pmatrix} \] 3. Compute the cross product \( \mathbf{b} \times \mathbf{c} \): \[ \mathbf{b} \times \mathbf{c} = \begin{pmatrix} b_2 c_3 - b_3 c_2 \\ b_3 c_1 - b_1 c_3 \\ b_1 c_2 - b_2 c_1 \end{pmatrix} = \begin{pmatrix} 5 \cdot 9 - 0 \cdot 2 \\ 0 \cdot 0 - (-8) \cdot 9 \\ -8 \cdot 2 - 5 \cdot 0 \end{pmatrix} = \begin{pmatrix} 45 \\ 72 \\ -16 \end{pmatrix} \] 4. Compute the dot product \( \mathbf{a} \cdot (\mathbf{b} \times \mathbf{c}) \): \[ \mathbf{a} \cdot (\mathbf{b} \times \mathbf{c}) = (-2)(45) + 4(72) + 8(-16) \] \[ = -90 + 288 - 128 = 70 \] 5. The volume of the parallelepiped is the absolute value of the triple scalar product: \[ V = |70| = 70 \] Thus, the volume of the parallelepiped is \( 70 \).