https://file.aiquickdraw.com/mathbot/file/172881714215797ni9d5i.jpg

We are tasked with finding the cross product of vectors **u** and **v**, and then checking if the result is orthogonal to both **u** and **v**. 

Given:
- **u** = \(\langle 0, 3, -4 \rangle\)
- **v** = \(\langle 3, 0, -3 \rangle\)

To compute the cross product **u × v**, we can use the following determinant formula:

\[
\mathbf{u} \times \mathbf{v} =
\begin{vmatrix}
\mathbf{i} & \mathbf{j} & \mathbf{k} \\
u_1 & u_2 & u_3 \\
v_1 & v_2 & v_3
\end{vmatrix}
\]

Substituting the values of **u** and **v**:

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

This expands to:

\[
\mathbf{i} \left( 3 \times -3 - (-4) \times 0 \right) 
- \mathbf{j} \left( 0 \times -3 - (-4) \times 3 \right) 
+ \mathbf{k} \left( 0 \times 0 - 3 \times 3 \right)
\]

Simplifying each term:

- For \(\mathbf{i}\): \(3 \times -3 = -9\), so the coefficient of \(\mathbf{i}\) is \(-9\).
- For \(\mathbf{j}\): \(-(-4 \times 3) = 12\), so the coefficient of \(\mathbf{j}\) is \(-12\).
- For \(\mathbf{k}\): \(-9\), so the coefficient of \(\mathbf{k}\) is \(-9\).

Thus:

\[
\mathbf{u} \times \mathbf{v} = \langle -9, -12, -9 \rangle
\]

### Checking orthogonality:
A vector is orthogonal to another if their dot product is zero.

1. **u · (u × v)**:
\[
\mathbf{u} \cdot \mathbf{u} \times \mathbf{v} = 0 \times -9 + 3 \times -12 + (-4) \times -9 = -36 + 36 = 0
\]

2. **v · (u × v)**:
\[
\mathbf{v} \cdot \mathbf{u} \times \mathbf{v} = 3 \times -9 + 0 \times -12 + (-3) \times -9 = -27 + 27 = 0
\]

Since both dot products are zero, the cross product is orthogonal to both **u** and **v**.

Would you like more details or further clarification?

### Related Questions:
1. What is the geometric interpretation of the cross product?
2. Can the magnitude of the cross product be calculated, and what does it represent?
3. How does the right-hand rule help in determining the direction of the cross product?
4. Can you calculate the angle between **u** and **v** from the cross product?
5. What happens when two vectors are parallel? What is their cross product?

### Tip:
Always check the orthogonality by computing dot products, as it confirms the correctness of the cross product computation.

Find u × v and check that it is orthogonal to both u and v. Given u = (0, 3, -4), v = (3, 0, -3).

Learn how to find the cross product of vectors u = (0, 3, -4) and v = (3, 0, -3), and verify if it is orthogonal to both u and v. This problem covers vector algebra, cross product, and orthogonality, suitable for high school students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation