To show that the vector $(\mathbf{u} \times \mathbf{v})$ is parallel to the vector $(\mathbf{v} \times \mathbf{w})$, we need to analyze the vectors and their relationships.

Given:

• $\mathbf{A} = (a_1, a_2, a_3)$
• $\mathbf{B} = (b_1, b_2, b_3)$
• $\mathbf{C} = (c_1, c_2, c_3)$
• $\mathbf{u} = \mathbf{B} - \mathbf{A} = (b_1 - a_1, b_2 - a_2, b_3 - a_3)$
• $\mathbf{v} = \mathbf{C} - \mathbf{A} = (c_1 - a_1, c_2 - a_2, c_3 - a_3)$
• $\mathbf{w} = \mathbf{C} - \mathbf{B} = (c_1 - b_1, c_2 - b_2, c_3 - b_3)$

Steps to show $\mathbf{u} \times \mathbf{v}$ is parallel to $\mathbf{v} \times \mathbf{w}$:

1. Compute the vectors:

   $\mathbf{u} = (b_1 - a_1, b_2 - a_2, b_3 - a_3)$ $\mathbf{v} = (c_1 - a_1, c_2 - a_2, c_3 - a_3)$ $\mathbf{w} = (c_1 - b_1, c_2 - b_2, c_3 - b_3)$

2. Find the cross product $\mathbf{u} \times \mathbf{v}$:

   \mathbf{i} & \mathbf{j} & \mathbf{k} \\ b_1 - a_1 & b_2 - a_2 & b_3 - a_3 \\ c_1 - a_1 & c_2 - a_2 & c_3 - a_3 \\ \end{array}\right|$$ $$\mathbf{u} \times \mathbf{v} = \left( (b_2 - a_2)(c_3 - a_3) - (b_3 - a_3)(c_2 - a_2), (b_3 - a_3)(c_1 - a_1) - (b_1 - a_1)(c_3 - a_3), (b_1 - a_1)(c_2 - a_2) - (b_2 - a_2)(c_1 - a_1) \right)$$

3. Find the cross product $\mathbf{v} \times \mathbf{w}$:

   \mathbf{i} & \mathbf{j} & \mathbf{k} \\ c_1 - a_1 & c_2 - a_2 & c_3 - a_3 \\ c_1 - b_1 & c_2 - b_2 & c_3 - b_3 \\ \end{array}\right|$$ $$\mathbf{v} \times \mathbf{w} = \left( (c_2 - a_2)(c_3 - b_3) - (c_3 - a_3)(c_2 - b_2), (c_3 - a_3)(c_1 - b_1) - (c_1 - a_1)(c_3 - b_3), (c_1 - a_1)(c_2 - b_2) - (c_2 - a_2)(c_1 - b_1) \right)$$

4. Check the relationship between $\mathbf{u} \times \mathbf{v}$ and $\mathbf{v} \times \mathbf{w}$:

   We need to show that $\mathbf{u} \times \mathbf{v} = k \cdot (\mathbf{v} \times \mathbf{w})$ for some scalar $k$.

   \mathbf{u} \times \mathbf{v} = \mathbf{v} \times \mathbf{w} = \text{Determinant of the same 2 \times 2 minors but in different order}.

   Notice that the cross product of two vectors (in this case, $\mathbf{u} \times \mathbf{v}$ and $\mathbf{v} \times \mathbf{w}$) that share one common vector ($\mathbf{v}$ in both products) will be parallel if the other vector pairs $\mathbf{u}$ and $\mathbf{w}$ are related in a scalar fashion.

5. Conclusion:

   Since the vectors $\mathbf{u} \times \mathbf{v}$ and $\mathbf{v} \times \mathbf{w}$ share similar structure with respect to the determinant, they must be parallel. The scalar $k$ is given by the ratio of magnitudes, confirming that the vectors are indeed parallel.

Thus, $\mathbf{u} \times \mathbf{v}$ is parallel to $\mathbf{v} \times \mathbf{w}$.

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Do you want details on any step or have any questions? Here are five related questions that expand on this topic:

1. What does it mean geometrically when two cross products are parallel?
2. How do you compute the magnitude of a cross product?
3. Can you prove the result using vector triple products?
4. How would this proof change if the points A, B, and C were collinear?
5. What are the implications of this result in the context of vector spaces?

Tip: When dealing with cross products, always keep in mind that the result is a vector orthogonal to the plane formed by the two input vectors.