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,
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
Problem 1RQ
Related questions
Question
![### 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.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F0fdef185-de8b-42f5-933e-8018dcf36541%2F68f0d835-5685-4a9f-9383-20fd8cd1da80%2Fbba5fzo_processed.png&w=3840&q=75)
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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