Question 10 Find the volume of the parallelepiped determined by the vectors \( \vec{a}=\langle 4,5,-1\rangle, \vec{b}=\langle 0,1,3\rangle \), \( \vec{c}=\langle 2,4,1\rangle \). Volume \( =\square \) pubic-units Submit Question Question 11

To find the volume of the parallelepiped determined by the vectors \( \vec{a} = \langle 4, 5, -1 \rangle \), \( \vec{b} = \langle 0, 1, 3 \rangle \), and \( \vec{c} = \langle 2, 4, 1 \rangle \), we can use the scalar triple product formula: \[ \text{Volume} = \left| \vec{a} \cdot (\vec{b} \times \vec{c}) \right| \] **Step 1: Compute the Cross Product \( \vec{b} \times \vec{c} \)** \[ \vec{b} = \langle 0, 1, 3 \rangle, \quad \vec{c} = \langle 2, 4, 1 \rangle \] \[ \vec{b} \times \vec{c} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ 0 & 1 & 3 \\ 2 & 4 & 1 \\ \end{vmatrix} = \mathbf{i}(1 \cdot 1 - 3 \cdot 4) - \mathbf{j}(0 \cdot 1 - 3 \cdot 2) + \mathbf{k}(0 \cdot 4 - 1 \cdot 2) \] \[ \vec{b} \times \vec{c} = \langle -11, 6, -2 \rangle \] **Step 2: Compute the Dot Product \( \vec{a} \cdot (\vec{b} \times \vec{c}) \)** \[ \vec{a} = \langle 4, 5, -1 \rangle, \quad \vec{b} \times \vec{c} = \langle -11, 6, -2 \rangle \] \[ \vec{a} \cdot (\vec{b} \times \vec{c}) = 4(-11) + 5(6) + (-1)(-2) = -44 + 30 + 2 = -12 \] **Step 3: Take the Absolute Value** \[ \text{Volume} = \left| -12 \right| = 12 \, \text{units}^3 \] **Final Answer:** \[ \text{Volume} = 12 \; \text{units}^3 \]