The identity \((u+v) \times (u-v) = 2(v \times u)\) is **not correct** as stated. Here's the detailed reasoning:

### Correct Cross Product Formula:
The cross product has the following distributive property:

\[
(u+v) \times (u-v) = u \times (u-v) + v \times (u-v)
\]

Expanding both terms, we get:

\[
= u \times u - u \times v + v \times u - v \times v
\]

Now, simplify:

- \(u \times u = 0\) (since the cross product of a vector with itself is zero),
- \(v \times v = 0\) (for the same reason),
- \(u \times v = -(v \times u)\) (since the cross product is anti-commutative).

Thus, the expression becomes:

\[
0 - u \times v + v \times u - 0
\]

Now, combine terms:

\[
= -(u \times v) + (v \times u)
\]

But, since \(-(u \times v) = v \times u\), we get:

\[
(u+v) \times (u-v) = 2(v \times u)
\]

So, the correct identity is:

\[
(u+v) \times (u-v) = 2(v \times u)
\]

### Conclusion:
Yes, the equation is true: \((u+v) \times (u-v) = 2(v \times u)\).

Would you like more details or have any questions?

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Here are 5 related questions for you to explore:

1. What is the distributive property of the cross product?
2. How does the cross product differ from the dot product in vector algebra?
3. Can you prove that \( u \times u = 0 \) for any vector \( u \)?
4. What is the geometric interpretation of the cross product?
5. How does the cross product change if \( u \) and \( v \) are parallel vectors?

**Tip**: The cross product is always perpendicular to the two vectors involved, and its magnitude depends on the sine of the angle between them.

Explore the proof of the cross product identity (u+v) × (u-v) = 2(v × u) using the distributive property and anti-commutative nature of the cross product. Learn about vector algebra concepts such as the cross product and how vectors interact geometrically. Suitable for undergraduate students.

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