Let's break down and solve each part of the problem step by step.

We are given three vectors $\mathbf{a}, \mathbf{b}, \mathbf{c} \in \mathbb{R}^3$ with $\mathbf{a} \neq \mathbf{b} \neq \mathbf{c} \neq 0$, and we are asked to determine whether the following two vector identities hold:

(i) $\mathbf{a} \times (\mathbf{b} \times \mathbf{c}) = (\mathbf{a} \times \mathbf{b}) + (\mathbf{a} \times \mathbf{c})$

This is a vector triple product identity involving the cross product. Let's explore if this equation holds:

Left-hand side (LHS): $\mathbf{a} \times (\mathbf{b} \times \mathbf{c})$

This expression is the vector triple product. The vector triple product identity states:

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

Thus, we can express the LHS as: $\mathbf{a} \times (\mathbf{b} \times \mathbf{c}) = (\mathbf{a} \cdot \mathbf{c}) \mathbf{b} - (\mathbf{a} \cdot \mathbf{b}) \mathbf{c}$

Right-hand side (RHS): $(\mathbf{a} \times \mathbf{b}) + (\mathbf{a} \times \mathbf{c})$

The RHS is just the sum of two separate cross products.

Conclusion for part (i):

Since the LHS involves a different form (scalar multiplication of vectors), and the RHS is a simple sum of cross products, the two sides are not equal. Thus, the identity does not hold.

(ii) $(\mathbf{a} \times \mathbf{b}) \times \mathbf{c} = (\mathbf{a} \cdot \mathbf{c}) \mathbf{b} - (\mathbf{b} \cdot \mathbf{c}) \mathbf{a}$

This is again a vector triple product identity.

Left-hand side (LHS): $(\mathbf{a} \times \mathbf{b}) \times \mathbf{c}$

By the vector triple product identity, we have:

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

Right-hand side (RHS): $(\mathbf{a} \cdot \mathbf{c}) \mathbf{b} - (\mathbf{b} \cdot \mathbf{c}) \mathbf{a}$

This is exactly the same expression as what we obtained for the LHS using the vector triple product identity.

Conclusion for part (ii):

Since both sides are equal, the identity holds true.

Final Answer:

• (i) The identity does not hold.
• (ii) The identity holds.

Would you like further details on any of these steps or explanations?

Related Questions:

1. What is the geometric interpretation of the vector cross product?
2. Can you explain the physical significance of the vector triple product identity?
3. How does the dot product relate to the cross product in these identities?
4. Why does the first identity fail to hold, while the second one does?
5. Could we generalize these identities to higher dimensions (e.g., $\mathbb{R}^n$)?

Tip:

When working with cross and dot products, always consider using established identities (like the vector triple product identity) to simplify the computations efficiently.