Find u ⨯ v, v ⨯ u, and v ⨯ v.
u	 = 	−3i + 2j
v	 = 	5i + 3j + 4k

To find the cross products \( \mathbf{u} \times \mathbf{v} \), \( \mathbf{v} \times \mathbf{u} \), and \( \mathbf{v} \times \mathbf{v} \), we will use the following vectors:

\[
\mathbf{u} = -3\mathbf{i} + 2\mathbf{j}
\]
\[
\mathbf{v} = 5\mathbf{i} + 3\mathbf{j} + 4\mathbf{k}
\]

### 1. \( \mathbf{u} \times \mathbf{v} \)

The cross product \( \mathbf{u} \times \mathbf{v} \) is calculated as:

\[
\mathbf{u} \times \mathbf{v} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ -3 & 2 & 0 \\ 5 & 3 & 4 \end{vmatrix}
\]

Expanding the determinant:

\[
\mathbf{u} \times \mathbf{v} = \mathbf{i} \left( 2 \times 4 - 0 \times 3 \right) - \mathbf{j} \left( -3 \times 4 - 0 \times 5 \right) + \mathbf{k} \left( -3 \times 3 - 2 \times 5 \right)
\]

\[
\mathbf{u} \times \mathbf{v} = \mathbf{i}(8) - \mathbf{j}(-12) + \mathbf{k}(-19)
\]

\[
\mathbf{u} \times \mathbf{v} = 8\mathbf{i} + 12\mathbf{j} - 19\mathbf{k}
\]

### 2. \( \mathbf{v} \times \mathbf{u} \)

The cross product \( \mathbf{v} \times \mathbf{u} \) is the opposite of \( \mathbf{u} \times \mathbf{v} \):

\[
\mathbf{v} \times \mathbf{u} = -(\mathbf{u} \times \mathbf{v})
\]

\[
\mathbf{v} \times \mathbf{u} = - \left( 8\mathbf{i} + 12\mathbf{j} - 19\mathbf{k} \right)
\]

\[
\mathbf{v} \times \mathbf{u} = -8\mathbf{i} - 12\mathbf{j} + 19\mathbf{k}
\]

### 3. \( \mathbf{v} \times \mathbf{v} \)

The cross product of any vector with itself is always the zero vector:

\[
\mathbf{v} \times \mathbf{v} = \mathbf{0}
\]

### Summary of Results

\[
\mathbf{u} \times \mathbf{v} = 8\mathbf{i} + 12\mathbf{j} - 19\mathbf{k}
\]
\[
\mathbf{v} \times \mathbf{u} = -8\mathbf{i} - 12\mathbf{j} + 19\mathbf{k}
\]
\[
\mathbf{v} \times \mathbf{v} = \mathbf{0}
\]

Would you like any details or have any questions?

Here are 5 related questions to consider:
1. How do you calculate the cross product in three dimensions?
2. What is the significance of the cross product being zero?
3. How does the cross product relate to the area of a parallelogram?
4. What is the difference between the dot product and the cross product?
5. How do you determine the direction of the cross product?

**Tip:** The magnitude of the cross product gives the area of the parallelogram formed by the two vectors.

Learn how to calculate cross products in 3D vectors using the vectors -3i + 2j and 5i + 3j + 4k. Understand the properties of cross products and their applications. Suitable for advanced high school students and anyone interested in vector mathematics.

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