To find the cosine of the angle between the vectors $u \times v$ and $u \times w$, we need to follow these steps:

Step 1: Calculate the cross products

• First, find the cross product $u \times v$.
• Then, find the cross product $u \times w$.

Step 2: Find the dot product of $u \times v$ and $u \times w$.

Step 3: Calculate the magnitudes of $u \times v$ and $u \times w$.

Step 4: Use the cosine formula:

$\cos \theta = \frac{(u \times v) \cdot (u \times w)}{|u \times v| \cdot |u \times w|}$

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Here are some related questions to expand your understanding:

1. How is the cross product of two vectors geometrically interpreted?
2. What is the dot product and how is it related to the cosine of the angle between two vectors?
3. How do you compute the magnitude of a vector?
4. Can the cross product result in a zero vector? Under what conditions?
5. What role does the cosine of an angle play in determining vector alignment?

Tip: The cross product of two vectors results in a vector that is perpendicular to both original vectors, forming the basis for many applications in physics and engineering.