The problem you've provided asks about two repeated vector products:

$\mathbf{i} \times (\mathbf{i} \times \mathbf{j}) \quad \text{and} \quad (\mathbf{i} \times \mathbf{i}) \times \mathbf{j}$

Part a) Are these two products equal?

1. $\mathbf{i} \times (\mathbf{i} \times \mathbf{j})$:

We can apply the vector triple product identity, which states: $\mathbf{a} \times (\mathbf{b} \times \mathbf{c}) = (\mathbf{a} \cdot \mathbf{c}) \mathbf{b} - (\mathbf{a} \cdot \mathbf{b}) \mathbf{c}$ Using $\mathbf{a} = \mathbf{i}$, $\mathbf{b} = \mathbf{i}$, and $\mathbf{c} = \mathbf{j}$, we get: $\mathbf{i} \times (\mathbf{i} \times \mathbf{j}) = (\mathbf{i} \cdot \mathbf{j}) \mathbf{i} - (\mathbf{i} \cdot \mathbf{i}) \mathbf{j}$ We know that: \mathbf{i} \cdot \mathbf{j} = 0 \quad \text{(since \mathbf{i}$and$\mathbf{j} are orthogonal unit vectors)} and $\mathbf{i} \cdot \mathbf{i} = 1$ Therefore: $\mathbf{i} \times (\mathbf{i} \times \mathbf{j}) = 0 \cdot \mathbf{i} - 1 \cdot \mathbf{j} = -\mathbf{j}$

2. $(\mathbf{i} \times \mathbf{i}) \times \mathbf{j}$:

The cross product of any vector with itself is zero: $\mathbf{i} \times \mathbf{i} = 0$ So: $(\mathbf{i} \times \mathbf{i}) \times \mathbf{j} = 0 \times \mathbf{j} = 0$

Thus, the two products are not equal. The first product is $-\mathbf{j}$, and the second product is 0.

Part b) Generalizing this result for repeated products:

From this example, we can see that if one of the cross products involves a vector crossed with itself (like $\mathbf{i} \times \mathbf{i}$), it will result in zero. Therefore, for repeated vector products of the form $(\mathbf{i} \times \mathbf{i}) \times \mathbf{v}$, the result will always be zero. This is a general rule: the cross product of any vector with itself is always zero.

On the other hand, in cases like $\mathbf{v} \times (\mathbf{v} \times \mathbf{w})$, we can use the vector triple product identity to simplify the expression.

Would you like further details or explanations on any part of this?

Here are some related questions to expand on this topic:

1. What is the geometric interpretation of the cross product?
2. How do you derive the vector triple product identity?
3. Can you apply the vector triple product identity to non-unit vectors?
4. What happens when the vectors in a cross product are not orthogonal?
5. How can you generalize this for non-orthogonal vector products?

Tip: The cross product of two vectors is a vector that is perpendicular to both, and its magnitude is related to the sine of the angle between them.