Compute each of the cross products:
ixj=? ixk=? jxk=?

The cross product of two standard unit vectors in 3D space (\(i, j, k\)) can be determined using the properties of the cross product. Here are the results:

1. **\(\mathbf{i} \times \mathbf{j}\):**
   \[
   \mathbf{i} \times \mathbf{j} = \mathbf{k}
   \]
   This is because the cross product of two perpendicular unit vectors results in the third unit vector that completes the right-handed coordinate system.

2. **\(\mathbf{i} \times \mathbf{k}\):**
   \[
   \mathbf{i} \times \mathbf{k} = -\mathbf{j}
   \]
   This is the result of the cross product between \(i\) and \(k\). It gives a vector that is perpendicular to both, but with the negative sign since the order matters (cross product is anti-commutative).

3. **\(\mathbf{j} \times \mathbf{k}\):**
   \[
   \mathbf{j} \times \mathbf{k} = \mathbf{i}
   \]
   This follows the same reasoning, resulting in a vector perpendicular to both \(j\) and \(k\), in the positive direction according to the right-hand rule.

To summarize:
\[
\mathbf{i} \times \mathbf{j} = \mathbf{k}, \quad \mathbf{i} \times \mathbf{k} = -\mathbf{j}, \quad \mathbf{j} \times \mathbf{k} = \mathbf{i}
\]

Would you like more details or have any questions?

### Related Questions:
1. What is the result of \(\mathbf{k} \times \mathbf{i}\)?
2. How does the right-hand rule help determine the direction of cross products?
3. What are the properties of the cross product in vector algebra?
4. Can cross products be computed in dimensions other than three?
5. How do you compute the cross product of two arbitrary vectors?

**Tip:** The cross product of vectors is useful for finding a vector perpendicular to a plane formed by two vectors.

Compute each of the cross products: ixj=? ixk=? jxk=?

Discover how to compute the cross products of standard unit vectors i, j, and k in 3D space. This problem covers essential concepts in vector algebra, including the right-hand rule and the anticommutativity property of the cross product. Suitable for students in Grades 10-12.

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