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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.

Consider the two repeated vector products i × (i × j) and (i × i) × j. a) Are these two products equal? b) Can you generalize this result for this type of repeated product?

Explore the solution to repeated vector products, including i × (i × j) and (i × i) × j. Learn how to use the vector triple product identity and the property that the cross product of a vector with itself is zero. This problem is suitable for advanced high school or undergraduate students.

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