Given the vectors $i$ and $j$, compute $(i\times j)\times k$ using the properties of cross products (not determinants). Which of the following is the result?

• A. $i$
• B. $-j$
• C. $j$
• D. $k$

Answer: (B)

Verified step by step guidance

1. Recall the basic properties of the cross product for the standard unit vectors \( \mathbf{i}, \mathbf{j}, \mathbf{k} \): \( \mathbf{i} \times \mathbf{j} = \mathbf{k} \), \( \mathbf{j} \times \mathbf{k} = \mathbf{i} \), and \( \mathbf{k} \times \mathbf{i} = \mathbf{j} \).
2. Start by computing the inner cross product \( (\mathbf{i} \times \mathbf{j}) \). Using the property above, this equals \( \mathbf{k} \).
3. Next, compute the cross product of the result with \( \mathbf{k} \), that is \( \mathbf{k} \times \mathbf{k} \).
4. Recall that the cross product of any vector with itself is the zero vector, so \( \mathbf{k} \times \mathbf{k} = \mathbf{0} \).
5. Therefore, the entire expression \( (\mathbf{i} \times \mathbf{j}) \times \mathbf{k} \) simplifies to \( \mathbf{0} \), which is not among the given options, so re-examine the problem or the properties used.