To show that the cross product $\mathbf{u} \times \mathbf{v}$ is parallel to $\mathbf{v} \times \mathbf{w}$, you need to demonstrate that these two vectors are scalar multiples of each other. Here’s how you can approach this problem step by step:

1. Define the vectors $\mathbf{u}$, $\mathbf{v}$, and $\mathbf{w}$:

Given the points $A = (a_1, a_2, a_3)$, $B = (b_1, b_2, b_3)$, and $C = (c_1, c_2, c_3)$, we define:

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

2. Express $\mathbf{w}$ in terms of $\mathbf{u}$ and $\mathbf{v}$:

Notice that: $\mathbf{w} = \mathbf{BC} = \mathbf{AC} - \mathbf{AB} = \mathbf{v} - \mathbf{u}$

3. Compute the cross products $\mathbf{u} \times \mathbf{v}$ and $\mathbf{v} \times \mathbf{w}$:

Now, the cross products are:

• $\mathbf{u} \times \mathbf{v}$
• $\mathbf{v} \times \mathbf{w} = \mathbf{v} \times (\mathbf{v} - \mathbf{u})$

Since the cross product is distributive over addition: $\mathbf{v} \times \mathbf{w} = \mathbf{v} \times \mathbf{v} - \mathbf{v} \times \mathbf{u}$

But, $\mathbf{v} \times \mathbf{v} = \mathbf{0}$ (because the cross product of any vector with itself is zero), so: $\mathbf{v} \times \mathbf{w} = -\mathbf{v} \times \mathbf{u}$

4. Show that the cross products are parallel:

From the above result, we have: $\mathbf{u} \times \mathbf{v} = -(\mathbf{v} \times \mathbf{u})$

Also: $\mathbf{v} \times \mathbf{w} = -(\mathbf{v} \times \mathbf{u})$

This implies: $\mathbf{u} \times \mathbf{v} = \mathbf{v} \times \mathbf{w} \text{ or } \mathbf{u} \times \mathbf{v} = -(\mathbf{v} \times \mathbf{w})$

In either case, $\mathbf{u} \times \mathbf{v}$ and $\mathbf{v} \times \mathbf{w}$ are scalar multiples of each other. Since they differ only by a sign, the two vectors are parallel.

Conclusion:

Thus, $\mathbf{u} \times \mathbf{v}$ is indeed parallel to $\mathbf{v} \times \mathbf{w}$, as they are scalar multiples of each other.

Would you like more details on any part of this explanation? Here are a few related questions you might find interesting:

1. How do you compute the magnitude of a cross product?
2. Why is the cross product of two vectors perpendicular to both vectors?
3. How do you determine the direction of the cross product?
4. What are some applications of cross products in physics?
5. How would the result change if we had three-dimensional vectors?

Tip: When two vectors are parallel, their cross product is either zero or another vector parallel to both, depending on whether they point in the same or opposite directions.