Given the vectors $i+j$ and $i-j$, what is the result of their cross product $\times$ using the properties of cross products (not determinants)?

• A. $2k$
• B. $0$
• C. $2j$
• D. $2i$

Answer: (B)

Verified step by step guidance

1. Identify the given vectors: \( \vec{A} = \hat{i} + \hat{j} \) and \( \vec{B} = \hat{i} - \hat{j} \).
2. Recall the properties of the cross product for unit vectors: \( \hat{i} \times \hat{i} = \vec{0} \), \( \hat{j} \times \hat{j} = \vec{0} \), \( \hat{i} \times \hat{j} = \hat{k} \), and \( \hat{j} \times \hat{i} = -\hat{k} \).
3. Use the distributive property of the cross product: \( \vec{A} \times \vec{B} = (\hat{i} + \hat{j}) \times (\hat{i} - \hat{j}) = \hat{i} \times \hat{i} - \hat{i} \times \hat{j} + \hat{j} \times \hat{i} - \hat{j} \times \hat{j} \).
4. Substitute the known cross products from step 2 into the expression: \( \vec{A} \times \vec{B} = \vec{0} - \hat{k} + (-\hat{k}) - \vec{0} \).
5. Combine like terms to find the final vector result of the cross product.