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To prove that the volume \( V \) of a parallelepiped whose sides are given by vectors \( \mathbf{a} \), \( \mathbf{b} \), and \( \mathbf{c} \) is:

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

we'll go through each step in the proof, explaining how the formula arises from vector operations.

### Step 1: Understanding the Volume of a Parallelepiped
A parallelepiped is a three-dimensional shape with six parallelogram faces. If three vectors \( \mathbf{a} \), \( \mathbf{b} \), and \( \mathbf{c} \) define three edges that meet at one vertex, then the volume \( V \) of this parallelepiped can be determined by the area of the base and the height.

1. **Base Area**: The base of the parallelepiped can be defined by vectors \( \mathbf{a} \) and \( \mathbf{b} \). The area of the parallelogram formed by these two vectors is given by the magnitude of their cross product \( |\mathbf{a} \times \mathbf{b}| \).

2. **Height**: The height is the perpendicular distance from the base to the opposite face, along the direction of \( \mathbf{c} \). This height is the component of \( \mathbf{c} \) along the direction perpendicular to the base, which we calculate using the dot product.

### Step 2: Computing the Volume Using the Cross Product and Dot Product
1. **Cross Product \( \mathbf{a} \times \mathbf{b} \)**: 
   The cross product \( \mathbf{a} \times \mathbf{b} \) produces a vector that is perpendicular to both \( \mathbf{a} \) and \( \mathbf{b} \). The magnitude \( |\mathbf{a} \times \mathbf{b}| \) represents the area of the parallelogram base.

2. **Dot Product \( (\mathbf{a} \times \mathbf{b}) \cdot \mathbf{c} \)**:
   To find the volume, we need the height of the parallelepiped in the direction of \( \mathbf{c} \) relative to the base. This is done by projecting \( \mathbf{c} \) onto the normal vector \( \mathbf{a} \times \mathbf{b} \), giving the component of \( \mathbf{c} \) in that direction.

3. The dot product \( (\mathbf{a} \times \mathbf{b}) \cdot \mathbf{c} \) thus represents the product of the area of the base \( |\mathbf{a} \times \mathbf{b}| \) and the height component in the direction of \( \mathbf{c} \). 

### Step 3: Taking the Absolute Value
Since volume is always a positive quantity, we take the absolute value of this result:

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

This completes the proof. The volume of the parallelepiped is determined by the absolute value of the scalar triple product, which geometrically represents the volume.

Would you like further clarification or more details on this proof?

### Related Questions
1. What are some geometric interpretations of the cross product in three dimensions?
2. How does the scalar triple product relate to the orientation of vectors in 3D space?
3. Can the volume of a parallelepiped be negative, and if so, what does that mean?
4. How would you find the volume of a parallelepiped if given coordinates instead of vectors?
5. What is the significance of the dot product in projecting vectors in physics and geometry?

### Tip
Remember, the scalar triple product \( (\mathbf{a} \times \mathbf{b}) \cdot \mathbf{c} \) is only zero if \( \mathbf{a} \), \( \mathbf{b} \), and \( \mathbf{c} \) are coplanar, indicating zero volume.

Prove that the volume V of a parallelepiped whose sides are given by the vectors a, b, and c, is V = |(a × b) · c|. Give a full proof and do not just quote Theorem 4.3.5.

Learn how to prove the volume formula for a parallelepiped defined by vectors a, b, and c, using the scalar triple product |(a × b) · c|. This problem includes key concepts in vector calculus, such as cross product and dot product, and provides a step-by-step proof suitable for undergraduate students.

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