https://file.aiquickdraw.com/mathbot/file/17266588988183ke72k6i.jpg

Here are the results for the given calculations:

1. **\( a - b \):**  
   \[
   a - b = (2, 6, -3)
   \]
2. **\( a \cdot b \) (dot product):**  
   \[
   a \cdot b = -7
   \]
3. **\( a \times b \) (cross product):**  
   \[
   a \times b = (0, -7, -14)
   \]
4. **\( |a \times b| \) (magnitude of cross product):**  
   \[
   |a \times b| \approx 15.65
   \]
5. **\( ab \) (outer product):**  
   \[
   ab = \begin{bmatrix} 3 & -12 & 6 \\ 2 & -8 & 4 \\ -1 & 4 & -2 \end{bmatrix}
   \]

Do you need more details or explanations? Here are 5 related questions to expand on this topic:

1. How is the cross product between two vectors geometrically interpreted?
2. What does the magnitude of the cross product represent?
3. How can the dot product be used to determine the angle between two vectors?
4. In which scenarios is the outer product of two vectors used?
5. What are the properties of the cross product in higher dimensions?

**Tip:** Remember, the cross product only applies to 3D vectors and results in a vector perpendicular to both input vectors.

Given a = 3e1 + 2e2 - e3, b = e1 - 4e2 + 2e3. Calculate: a - b, a · b, a × b, |a × b|, ab.

Solve the vector problem involving vector subtraction, dot product, cross product, and the outer product for vectors a and b. Detailed step-by-step solution provided, suitable for undergraduate students.

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