Find the volume of the parallelepiped with one vertex (𝟐, 𝟐, 𝟎), and adjacent vertices are (𝟔, 𝟒, −𝟐), (𝟑, −𝟒, −𝟏), and (𝟕, 𝟑, −𝟐).

To find the volume of a parallelepiped with one vertex and three adjacent vertices, we need to calculate the scalar triple product of the vectors formed by these adjacent vertices.

Given:
- One vertex at \( (2, 2, 0) \)
- Adjacent vertices at \( (6, 4, -2) \), \( (3, -4, -1) \), and \( (7, 3, -2) \)

First, we'll form vectors based on these points:

### Step 1: Calculate vectors from the reference vertex \( (2, 2, 0) \)

\[
\mathbf{v_1} = (6, 4, -2) - (2, 2, 0) = (6 - 2, 4 - 2, -2 - 0) = (4, 2, -2)
\]
\[
\mathbf{v_2} = (3, -4, -1) - (2, 2, 0) = (3 - 2, -4 - 2, -1 - 0) = (1, -6, -1)
\]
\[
\mathbf{v_3} = (7, 3, -2) - (2, 2, 0) = (7 - 2, 3 - 2, -2 - 0) = (5, 1, -2)
\]

### Step 2: Calculate the scalar triple product

The volume \( V \) of the parallelepiped is the absolute value of the scalar triple product of vectors \( \mathbf{v_1}, \mathbf{v_2}, \mathbf{v_3} \). This can be calculated as:

\[
V = |\mathbf{v_1} \cdot (\mathbf{v_2} \times \mathbf{v_3})|
\]

### Step 3: Compute the cross product \( \mathbf{v_2} \times \mathbf{v_3} \)

\[
\mathbf{v_2} \times \mathbf{v_3} = \begin{vmatrix}
\mathbf{i} & \mathbf{j} & \mathbf{k} \\
1 & -6 & -1 \\
5 & 1 & -2
\end{vmatrix}
\]

\[
\mathbf{v_2} \times \mathbf{v_3} = \mathbf{i} \left( (-6)(-2) - (-1)(1) \right) - \mathbf{j} \left( (1)(-2) - (-1)(5) \right) + \mathbf{k} \left( (1)(1) - (-6)(5) \right)
\]

\[
= \mathbf{i} (12 + 1) - \mathbf{j} (-2 + 5) + \mathbf{k} (1 + 30)
\]
\[
= \mathbf{i} (13) - \mathbf{j} (3) + \mathbf{k} (31)
\]
\[
\mathbf{v_2} \times \mathbf{v_3} = (13, -3, 31)
\]

### Step 4: Compute the dot product \( \mathbf{v_1} \cdot (\mathbf{v_2} \times \mathbf{v_3}) \)

\[
\mathbf{v_1} \cdot (13, -3, 31) = (4)(13) + (2)(-3) + (-2)(31)
\]
\[
= 52 - 6 - 62
\]
\[
= -16
\]

### Step 5: Calculate the volume

The volume is the absolute value of the scalar triple product:

\[
V = |-16| = 16
\]

Thus, the volume of the parallelepiped is \( \boxed{16} \) cubic units.

Would you like more details on any of the steps, or have any questions?

Here are 5 related questions to expand on this concept:
1. How do you compute the cross product of two vectors in 3D space?
2. What is the geometric interpretation of the scalar triple product?
3. How would the volume change if the vectors formed a non-rectangular parallelepiped?
4. Can the volume be negative, and what does that signify geometrically?
5. How do you find the area of a parallelogram formed by two vectors?

**Tip:** The scalar triple product can also indicate whether three vectors form a right-handed or left-handed system depending on the sign.

Find the volume of the parallelepiped with one vertex (2, 2, 0), and adjacent vertices are (6, 4, −2), (3, −4, −1), and (7, 3, −2).

Learn how to calculate the volume of a parallelepiped using the scalar triple product and cross product. The problem involves vector calculus and geometry concepts, with a step-by-step solution. Suitable for undergraduate-level students studying vector calculus.

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