To find the volume of the parallelepiped formed by the vectors t, u, and v, we use the scalar triple product. The formula for the volume is:

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

Where:

• $\mathbf{t} = -4\hat{i} + \hat{j} + 3\hat{k}$
• $\mathbf{u} = 5\hat{i} + 7\hat{j} - 6\hat{k}$
• $\mathbf{v} = 3\hat{i} - 2\hat{j} - 5\hat{k}$

Step 1: Calculate the cross product $\mathbf{u} \times \mathbf{v}$

The cross product of two vectors $\mathbf{a} = a_1\hat{i} + a_2\hat{j} + a_3\hat{k}$ and $\mathbf{b} = b_1\hat{i} + b_2\hat{j} + b_3\hat{k}$ is given by the determinant of the matrix:

\hat{i} & \hat{j} & \hat{k} \\ a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \end{vmatrix}$$ For $$\mathbf{u} = 5\hat{i} + 7\hat{j} - 6\hat{k}$$ and $$\mathbf{v} = 3\hat{i} - 2\hat{j} - 5\hat{k}$$, the cross product is: $$\mathbf{u} \times \mathbf{v} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ 5 & 7 & -6 \\ 3 & -2 & -5 \end{vmatrix}$$ We compute the determinant: $$= \hat{i} \begin{vmatrix} 7 & -6 \\ -2 & -5 \end{vmatrix} - \hat{j} \begin{vmatrix} 5 & -6 \\ 3 & -5 \end{vmatrix} + \hat{k} \begin{vmatrix} 5 & 7 \\ 3 & -2 \end{vmatrix}$$ \[ = \hat{i} [(7)(-5) - (-6)(-2)] - \hat{j} [(5)(-5) - (-6)(3)] + \hat{k} [(5)(-2) - (7)(3)] \] \[ = \hat{i} [-35 - 12] - \hat{j} [-25 + 18] + \hat{k} [-10 - 21] \] $$= \hat{i} (-47) - \hat{j} (-7) + \hat{k} (-31)$$ $$= -47\hat{i} + 7\hat{j} - 31\hat{k}$$ Thus, the cross product $$\mathbf{u} \times \mathbf{v} = -47\hat{i} + 7\hat{j} - 31\hat{k}$$. ### Step 2: Compute the dot product $$\mathbf{t} \cdot (\mathbf{u} \times \mathbf{v})$$ Now, take the dot product of $$\mathbf{t} = -4\hat{i} + \hat{j} + 3\hat{k}$$ with $$\mathbf{u} \times \mathbf{v} = -47\hat{i} + 7\hat{j} - 31\hat{k}$$: $$\mathbf{t} \cdot (\mathbf{u} \times \mathbf{v}) = (-4)(-47) + (1)(7) + (3)(-31)$$ $$= 188 + 7 - 93$$ $$= 102$$ ### Step 3: Compute the volume The volume of the parallelepiped is the absolute value of the scalar triple product: $$V = |102| = 102$$ Thus, the volume of the parallelepiped is **102 cubic units**. --- Would you like further details on any of the steps or concepts used? Here are some related questions for your understanding: 1. How do the components of the scalar triple product represent geometric properties of a parallelepiped? 2. How do you compute the cross product of two vectors geometrically? 3. What are the differences between the cross product and dot product in vector calculations? 4. What happens to the volume of a parallelepiped if the vectors are linearly dependent? 5. Can the scalar triple product be used to find the volume of other shapes, such as tetrahedra? **Tip:** When calculating the cross product, ensure that you correctly apply the sign changes for each of the unit vectors $$\hat{i}$$, $$\hat{j}$$, and $$\hat{k}$$ in the determinant.