https://file.aiquickdraw.com/mathbot/file/1730302159071ge3n05b4.jpg

To find the cross product \( \mathbf{u} \times \mathbf{v} \) of the given vectors:

\[
\mathbf{u} = \begin{bmatrix} 3 \\ 0 \\ -1 \end{bmatrix} \quad \text{and} \quad \mathbf{v} = \begin{bmatrix} 4 \\ 0 \\ -2 \end{bmatrix},
\]

we use the formula for the cross product of two 3-dimensional vectors:
\[
\mathbf{u} \times \mathbf{v} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ 3 & 0 & -1 \\ 4 & 0 & -2 \end{vmatrix}.
\]

This expands to:
\[
\mathbf{u} \times \mathbf{v} = \mathbf{i} \begin{vmatrix} 0 & -1 \\ 0 & -2 \end{vmatrix} - \mathbf{j} \begin{vmatrix} 3 & -1 \\ 4 & -2 \end{vmatrix} + \mathbf{k} \begin{vmatrix} 3 & 0 \\ 4 & 0 \end{vmatrix}.
\]

Now we compute each minor determinant:

1. **For \( \mathbf{i} \)-component:**
   \[
   \begin{vmatrix} 0 & -1 \\ 0 & -2 \end{vmatrix} = (0)(-2) - (0)(-1) = 0.
   \]

2. **For \( \mathbf{j} \)-component:**
   \[
   \begin{vmatrix} 3 & -1 \\ 4 & -2 \end{vmatrix} = (3)(-2) - (-1)(4) = -6 + 4 = -2.
   \]

3. **For \( \mathbf{k} \)-component:**
   \[
   \begin{vmatrix} 3 & 0 \\ 4 & 0 \end{vmatrix} = (3)(0) - (0)(4) = 0.
   \]

Thus, the cross product is:
\[
\mathbf{u} \times \mathbf{v} = 0 \mathbf{i} - (-2) \mathbf{j} + 0 \mathbf{k} = \begin{bmatrix} 0 \\ 2 \\ 0 \end{bmatrix}.
\]

### Final Answer:
\[
\mathbf{u} \times \mathbf{v} = \begin{bmatrix} 0 \\ 2 \\ 0 \end{bmatrix}.
\]

### Direction of the Cross Product
The direction of the cross product is along the positive \( y \)-axis, as the resulting vector points in the \( \mathbf{j} \) (or \( y \)-axis) direction.

Would you like further clarification on any step, or is there another concept here you'd like to explore? 

### Related Questions
1. How do you find the magnitude of a cross product vector?
2. Why is the cross product of two parallel vectors zero?
3. How does the right-hand rule apply to determine the direction of the cross product?
4. What are some physical applications of the cross product in physics?
5. What is the difference between the cross product and the dot product?

### Quick Tip
For 3D vectors, remember that the cross product is always perpendicular to the plane formed by the original two vectors.

You are given the co-planar vectors u = [3, 0, -1] and v = [4, 0, -2]. Calculate the vector product u x v.

Learn how to calculate the cross product of vectors u = [3, 0, -1] and v = [4, 0, -2] using the cross product formula and right-hand rule. This step-by-step solution covers key linear algebra concepts and is suitable for students in Grades 11-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation