The volume $V$ of a parallelepiped defined by three vectors $\mathbf{t}, \mathbf{u},$ and $\mathbf{v}$ is given by the absolute value of the scalar triple product:

$V = \left| \mathbf{t} \cdot (\mathbf{u} \times \mathbf{v}) \right|$

Step 1: Compute the Cross Product $\mathbf{u} \times \mathbf{v}$

Given: $\mathbf{t} = (-4i + j + 3k), \quad \mathbf{u} = (5i + 7j - 6k), \quad \mathbf{v} = (3i - 2j - 5k)$

The cross product $\mathbf{u} \times \mathbf{v}$ is given by:

\begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ 5 & 7 & -6 \\ 3 & -2 & -5 \end{vmatrix}$$ Expanding along the first row: $$\mathbf{u} \times \mathbf{v} = \mathbf{i} \begin{vmatrix} 7 & -6 \\ -2 & -5 \end{vmatrix} - \mathbf{j} \begin{vmatrix} 5 & -6 \\ 3 & -5 \end{vmatrix} + \mathbf{k} \begin{vmatrix} 5 & 7 \\ 3 & -2 \end{vmatrix}$$ Computing the determinants: $$\begin{vmatrix} 7 & -6 \\ -2 & -5 \end{vmatrix} = (7)(-5) - (-6)(-2) = -35 - 12 = -47$$ $$\begin{vmatrix} 5 & -6 \\ 3 & -5 \end{vmatrix} = (5)(-5) - (-6)(3) = -25 + 18 = -7$$ $$\begin{vmatrix} 5 & 7 \\ 3 & -2 \end{vmatrix} = (5)(-2) - (7)(3) = -10 - 21 = -31$$ Thus, $$\mathbf{u} \times \mathbf{v} = (-47i + 7j - 31k)$$ ### Step 2: Compute the Scalar Triple Product $$\mathbf{t} \cdot (\mathbf{u} \times \mathbf{v})$$ $$(-4i + j + 3k) \cdot (-47i + 7j - 31k)$$ $$= (-4)(-47) + (1)(7) + (3)(-31)$$ $$= 188 + 7 - 93 = 102$$ ### Step 3: Take the Absolute Value $$V = |102| = 102$$ ### Final Answer: $$\boxed{102}$$ Would you like a step-by-step breakdown for any part? Here are five related questions: 1. How do you find the volume of a parallelepiped given three edge vectors? 2. What is the geometric interpretation of the scalar triple product? 3. How do you compute the cross product of two vectors? 4. How do you determine if three vectors form a parallelepiped with zero volume? 5. What is the difference between a parallelepiped and a rectangular prism in vector terms? **Tip:** The absolute value in the scalar triple product ensures the volume is always non-negative! 🚀