Which of the following vectors is orthogonal to the plane 3(x - 7) + (y + 8) - 5z = 15? O <7,-8, 0> O <- 7, 8, 0> O <3, 1, - 5> <- - 3, -1, 5>
Which of the following vectors is orthogonal to the plane 3(x - 7) + (y + 8) - 5z = 15? O <7,-8, 0> O <- 7, 8, 0> O <3, 1, - 5> <- - 3, -1, 5>
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
![### Question:
**Which of the following vectors is orthogonal to the plane 3(x - 7) + (y + 8) - 5z = 15?**
### Answer Choices:
- ☐ \( \langle 7, -8, 0 \rangle \)
- ☐ \( \langle -7, 8, 0 \rangle \)
- ☐ \( \langle 3, 1, -5 \rangle \)
- ☐ \( \langle -3, -1, 5 \rangle \)
- ☐ None of these
### Explanation:
To determine which of the given vectors is orthogonal to the plane, we need to check if any of the vectors match the normal vector of the plane. The equation of the plane given is in the form:
\[ 3(x - 7) + (y + 8) - 5z = 15 \]
By expanding and rewriting this equation, we get:
\[ 3x - 21 + y + 8 - 5z = 15 \implies 3x + y - 5z - 13 = 0 \]
The coefficients of \( x \), \( y \), and \( z \) in the equation represent the components of the normal vector to the plane. Thus, the normal vector is:
\[ \langle 3, 1, -5 \rangle \]
Therefore, the vector orthogonal to the plane is:
**\( \langle 3, 1, -5 \rangle \)**
Hence, the correct answer is the third option.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F55815b3c-2575-4201-a9dc-5a4b41a14c1d%2F30486b29-09f3-4371-ada1-9871dc65b284%2F0ouxwre_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Question:
**Which of the following vectors is orthogonal to the plane 3(x - 7) + (y + 8) - 5z = 15?**
### Answer Choices:
- ☐ \( \langle 7, -8, 0 \rangle \)
- ☐ \( \langle -7, 8, 0 \rangle \)
- ☐ \( \langle 3, 1, -5 \rangle \)
- ☐ \( \langle -3, -1, 5 \rangle \)
- ☐ None of these
### Explanation:
To determine which of the given vectors is orthogonal to the plane, we need to check if any of the vectors match the normal vector of the plane. The equation of the plane given is in the form:
\[ 3(x - 7) + (y + 8) - 5z = 15 \]
By expanding and rewriting this equation, we get:
\[ 3x - 21 + y + 8 - 5z = 15 \implies 3x + y - 5z - 13 = 0 \]
The coefficients of \( x \), \( y \), and \( z \) in the equation represent the components of the normal vector to the plane. Thus, the normal vector is:
\[ \langle 3, 1, -5 \rangle \]
Therefore, the vector orthogonal to the plane is:
**\( \langle 3, 1, -5 \rangle \)**
Hence, the correct answer is the third option.
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 2 steps
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
