Suppose A= (a1,a2,a3), B = (b1,b2,b3), and C = (c1,c2,c3) are three non-colinear points. Let u, v, and w be the vectors defined as u = A to B, v = A to C, and w=B to C. Show that (u)(v) is parallel to (v)(w).

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 \(\mathbf{u} \times \mathbf{v}\):**

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

3. **Find the cross product \(\mathbf{v} \times \mathbf{w}\):**

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

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.

Learn how to prove that the cross product of vectors u and v is parallel to the cross product of vectors v and w using vector analysis. This problem involves advanced concepts in vectors and is suitable for advanced high school students and beyond.

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