Let u= (9, 4, 3) and 3). Find a nonzero vector orthogonal to both u and v. Enter your answer as a vector using < and > as enclosing brackets. = (4, 6,

Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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### Problem Statement

Let **u** = ⟨9, 4, 3⟩ and **v** = ⟨4, 6,  − 3⟩.

Find a nonzero vector orthogonal to both **u** and **v**. Enter your answer as a vector using `<` and `>` as enclosing brackets.

---

To solve the problem, we will utilize the cross product of the vectors **u** and **v**. The result will be a vector orthogonal to both given vectors.

The cross product **u** × **v** is calculated using the determinant of a matrix:
```
         |   i     j     k   |
         |  9     4     3   |
         |  4     6    -3   |
```
Expanding this determinant:
- **i** component: (4 * (-3)) - (6 * 3) = -12 - 18 = -30
- **j** component: - [(9 * (-3)) - (4 * 3)] = - [-27 - 12] = 39
- **k** component: (9 * 6) - (4 * 4) = 54 - 16 = 38

Thus, the orthogonal vector is ⟨-30, 39, 38⟩.

So, the nonzero vector orthogonal to both **u** and **v** is:
```
< -30, 39, 38 >
```
Enter the answer in the provided text box as indicated.
Transcribed Image Text:### Problem Statement Let **u** = ⟨9, 4, 3⟩ and **v** = ⟨4, 6,  − 3⟩. Find a nonzero vector orthogonal to both **u** and **v**. Enter your answer as a vector using `<` and `>` as enclosing brackets. --- To solve the problem, we will utilize the cross product of the vectors **u** and **v**. The result will be a vector orthogonal to both given vectors. The cross product **u** × **v** is calculated using the determinant of a matrix: ``` | i j k | | 9 4 3 | | 4 6 -3 | ``` Expanding this determinant: - **i** component: (4 * (-3)) - (6 * 3) = -12 - 18 = -30 - **j** component: - [(9 * (-3)) - (4 * 3)] = - [-27 - 12] = 39 - **k** component: (9 * 6) - (4 * 4) = 54 - 16 = 38 Thus, the orthogonal vector is ⟨-30, 39, 38⟩. So, the nonzero vector orthogonal to both **u** and **v** is: ``` < -30, 39, 38 > ``` Enter the answer in the provided text box as indicated.
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