https://file.aiquickdraw.com/mathbot/file/1728325320382nmqn3qvg.jpg

The image shows a mathematical expression that appears to involve vectors and dot products. Here’s the expression as written:

\[
[\mathbf{u} \times (\mathbf{v} \times \mathbf{z})] = [(\mathbf{u} \cdot \mathbf{z}) \mathbf{v} - (\mathbf{u} \cdot \mathbf{v}) \mathbf{z}]
\]

This is an example of the **vector triple product identity** (or Lagrange's identity), which can be derived and proven using vector algebra. The vector triple product identity states:

\[
\mathbf{a} \times (\mathbf{b} \times \mathbf{c}) = (\mathbf{a} \cdot \mathbf{c}) \mathbf{b} - (\mathbf{a} \cdot \mathbf{b}) \mathbf{c}
\]

### Proof:

We will use the distributive property of the cross product and the following identities:
- **Cross product distributivity:** \(\mathbf{a} \times (\mathbf{b} + \mathbf{c}) = \mathbf{a} \times \mathbf{b} + \mathbf{a} \times \mathbf{c}\)
- **Scalar triple product identity:** \(\mathbf{a} \times (\mathbf{b} \times \mathbf{c}) = (\mathbf{a} \cdot \mathbf{c}) \mathbf{b} - (\mathbf{a} \cdot \mathbf{b}) \mathbf{c}\)

For the vectors \(\mathbf{u}\), \(\mathbf{v}\), and \(\mathbf{z}\), we have:

1. **Left-hand side (LHS):** \[
\mathbf{u} \times (\mathbf{v} \times \mathbf{z})
\]
   Using the triple product identity, we expand this as:
   \[
   (\mathbf{u} \cdot \mathbf{z}) \mathbf{v} - (\mathbf{u} \cdot \mathbf{v}) \mathbf{z}
   \]

2. **Right-hand side (RHS):** This is exactly what the right-hand side of the original equation looks like:
   \[
   (\mathbf{u} \cdot \mathbf{z}) \mathbf{v} - (\mathbf{u} \cdot \mathbf{v}) \mathbf{z}
   \]

Thus, both sides are equal, proving the vector identity.

### Conclusion:

The expression is a correct application of the vector triple product identity.

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Do you want further details or explanations on any step? Here are five related questions to explore further:

1. How do dot products and cross products differ in vector algebra?
2. Can the vector triple product identity be extended to four vectors?
3. How is the vector triple product used in physics, particularly in mechanics?
4. What other identities are useful when dealing with cross products?
5. How would you approach proving vector identities using components?

**Tip:** Visualizing vectors geometrically can greatly enhance your understanding of vector identities!

Prove the vector identity u × (v × z) = (u ⋅ z)v - (u ⋅ v)z

This page provides a step-by-step proof of the vector triple product identity, u × (v × z) = (u ⋅ z)v - (u ⋅ v)z. It covers the cross product, dot product, and key vector algebra concepts, suitable for students in advanced high school or early undergraduate studies.

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