Evaluate (vector a cross vector b) dot (vector a - vector b)

Let's evaluate the expression \((\mathbf{a} \times \mathbf{b}) \cdot (\mathbf{a} - \mathbf{b})\).

Given:
\[
\mathbf{a}, \mathbf{b} \text{ are vectors, }
\]

### Step 1: Apply the distributive property
We can expand the dot product using the distributive property:
\[
(\mathbf{a} \times \mathbf{b}) \cdot (\mathbf{a} - \mathbf{b}) = (\mathbf{a} \times \mathbf{b}) \cdot \mathbf{a} - (\mathbf{a} \times \mathbf{b}) \cdot \mathbf{b}
\]

### Step 2: Evaluate each term
1. **First Term: \((\mathbf{a} \times \mathbf{b}) \cdot \mathbf{a}\)**

   The result of the cross product \(\mathbf{a} \times \mathbf{b}\) is a vector that is perpendicular to both \(\mathbf{a}\) and \(\mathbf{b}\). Since the dot product of a vector with a perpendicular vector is zero:
   \[
   (\mathbf{a} \times \mathbf{b}) \cdot \mathbf{a} = 0
   \]

2. **Second Term: \((\mathbf{a} \times \mathbf{b}) \cdot \mathbf{b}\)**

   Similarly, since \(\mathbf{a} \times \mathbf{b}\) is perpendicular to \(\mathbf{b}\):
   \[
   (\mathbf{a} \times \mathbf{b}) \cdot \mathbf{b} = 0
   \]

### Step 3: Combine results
Substituting the values from the previous steps, we get:
\[
(\mathbf{a} \times \mathbf{b}) \cdot (\mathbf{a} - \mathbf{b}) = 0 - 0 = 0
\]

### Final Answer
\[
(\mathbf{a} \times \mathbf{b}) \cdot (\mathbf{a} - \mathbf{b}) = 0
\]

Would you like any more details or have any further questions? Here are some related questions:

1. What is the geometric interpretation of a cross product?
2. How do you calculate the magnitude of a cross product?
3. What are the properties of the dot product of two vectors?
4. How do vector triple products work?
5. Can you explain the distributive property for dot and cross products?

**Tip:** Always remember that the cross product of two vectors is perpendicular to both vectors, which can simplify many dot product calculations.

Understand how to evaluate the expression (vector a cross vector b) dot (vector a - vector b) with a detailed step-by-step solution. This problem involves vector algebra, cross product properties, and dot product applications. Suitable for advanced high school students and college-level learners.

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