Determine the volume of a parallelepiped that has as edges the vectors 
a=<2,4,5>, b=<0,-7,1> and c=<6,-8,2

The volume \( V \) of a parallelepiped formed by vectors \( \mathbf{a}, \mathbf{b}, \mathbf{c} \) is given by the absolute value of the scalar triple product:

\[
V = |\mathbf{a} \cdot (\mathbf{b} \times \mathbf{c})|
\]

Let's calculate this step by step.

### Step 1: Compute the cross product \( \mathbf{b} \times \mathbf{c} \)
Given:
\[
\mathbf{b} = \langle 0, -7, 1 \rangle, \quad \mathbf{c} = \langle 6, -8, 2 \rangle
\]

The formula for the cross product \( \mathbf{b} \times \mathbf{c} \) is:

\[
\mathbf{b} \times \mathbf{c} = \begin{vmatrix}
\mathbf{i} & \mathbf{j} & \mathbf{k} \\
0 & -7 & 1 \\
6 & -8 & 2
\end{vmatrix}
\]

This results in:

\[
\mathbf{b} \times \mathbf{c} = \mathbf{i} \begin{vmatrix} -7 & 1 \\ -8 & 2 \end{vmatrix} - \mathbf{j} \begin{vmatrix} 0 & 1 \\ 6 & 2 \end{vmatrix} + \mathbf{k} \begin{vmatrix} 0 & -7 \\ 6 & -8 \end{vmatrix}
\]

\[
= \mathbf{i} \left((-7)(2) - (-8)(1)\right) - \mathbf{j} \left((0)(2) - (1)(6)\right) + \mathbf{k} \left((0)(-8) - (-7)(6)\right)
\]

\[
= \mathbf{i} (-14 + 8) - \mathbf{j} (0 - 6) + \mathbf{k} (0 + 42)
\]

\[
= \mathbf{i} (-6) - \mathbf{j} (-6) + \mathbf{k} (42)
\]

\[
= \langle -6, 6, 42 \rangle
\]

### Step 2: Compute the dot product \( \mathbf{a} \cdot (\mathbf{b} \times \mathbf{c}) \)
Now, we take the dot product of \( \mathbf{a} = \langle 2, 4, 5 \rangle \) with \( \mathbf{b} \times \mathbf{c} = \langle -6, 6, 42 \rangle \):

\[
\mathbf{a} \cdot (\mathbf{b} \times \mathbf{c}) = (2)(-6) + (4)(6) + (5)(42)
\]

\[
= -12 + 24 + 210 = 222
\]

### Step 3: Compute the volume
The volume is the absolute value of this dot product:

\[
V = |222| = 222
\]

Thus, the volume of the parallelepiped is \( 222 \, \text{cubic units} \).

Would you like more details or have any questions?

Here are 5 related questions to expand on this concept:
1. How do you geometrically interpret the scalar triple product in three dimensions?
2. Can the volume of a parallelepiped ever be negative?
3. How is the cross product of two vectors related to the area of a parallelogram?
4. What would happen to the volume if two of the vectors were parallel?
5. How can the scalar triple product be useful in physics, particularly in mechanics?

**Tip**: The scalar triple product gives both the volume and the orientation (sign) of the parallelepiped.

Determine the volume of a parallelepiped that has as edges the vectors a=<2,4,5>, b=<0,-7,1>, and c=<6,-8,2>.

Learn how to calculate the volume of a parallelepiped with vectors a=<2,4,5>, b=<0,-7,1>, and c=<6,-8,2> using the scalar triple product. This solution covers key concepts such as the cross product, dot product, and vector calculus. Suitable for Grades 10-12 and college-level students.

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