Show that the vector-valued function shown below describes the motion of a particle moving in a circle of radius 1 centered at the point (5,3,1) and lying in the plane x+y-2z=6 1 1 1 r(t) = (5i +3j+k) + cost + sint i+ First write the three parametric equations for x, y, and z. What is the parametric equation for x? X (Type an exact answer, using radicals as needed.)
Show that the vector-valued function shown below describes the motion of a particle moving in a circle of radius 1 centered at the point (5,3,1) and lying in the plane x+y-2z=6 1 1 1 r(t) = (5i +3j+k) + cost + sint i+ First write the three parametric equations for x, y, and z. What is the parametric equation for x? X (Type an exact answer, using radicals as needed.)
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![### Vector-Valued Functions: Describing Particle Motion
#### Problem Statement
Show that the vector-valued function shown below describes the motion of a particle moving in a circle of radius 1 centered at the point (5, 3, 1) and lying in the plane given by the equation \(x + y - 2z = 6\).
\[
r(t) = (5i + 3j + k) + \cos(t)
\left( \frac{1}{\sqrt{2}}i - \frac{1}{\sqrt{2}}j \right)
+ \sin(t) \left( \frac{1}{\sqrt{3}}i + \frac{1}{\sqrt{3}}j + \frac{1}{\sqrt{3}}k \right)
\]
---
#### Task
First write the three parametric equations for \(x\), \(y\), and \(z\). What is the parametric equation for \(x\)?
\( x = \_\_\_\_\_\_\_\_\_\_\ \)
(Type an exact answer, using radicals as needed.)
---
#### Explanation
To begin solving this problem, we need to express the vector-valued function \(r(t)\) in its component form and derive the parametric equations for \(x\), \(y\), and \(z\). The vector-valued function \(r(t)\) is given in terms of \(i\), \(j\), and \(k\), which represent the unit vectors along the coordinate axes.
1. **Vector Representation**:
\[
r(t) = (5i + 3j + k)
+ \cos(t) \left( \frac{1}{\sqrt{2}}i - \frac{1}{\sqrt{2}}j \right)
+ \sin(t) \left( \frac{1}{\sqrt{3}}i + \frac{1}{\sqrt{3}}j + \frac{1}{\sqrt{3}}k \right)
\]
2. **Component Form**:
\[
r(t) = 5 + \cos(t) \left( \frac{1}{\sqrt{2}} \right) + \sin(t) \left( \frac{1}{\sqrt{3}} \right)i
+ 3](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F41a1af84-f79e-495e-9bf6-e155c46aa5b8%2F1591e6f7-d1e8-49da-9972-383ece9c8e74%2F5qf0lpr_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Vector-Valued Functions: Describing Particle Motion
#### Problem Statement
Show that the vector-valued function shown below describes the motion of a particle moving in a circle of radius 1 centered at the point (5, 3, 1) and lying in the plane given by the equation \(x + y - 2z = 6\).
\[
r(t) = (5i + 3j + k) + \cos(t)
\left( \frac{1}{\sqrt{2}}i - \frac{1}{\sqrt{2}}j \right)
+ \sin(t) \left( \frac{1}{\sqrt{3}}i + \frac{1}{\sqrt{3}}j + \frac{1}{\sqrt{3}}k \right)
\]
---
#### Task
First write the three parametric equations for \(x\), \(y\), and \(z\). What is the parametric equation for \(x\)?
\( x = \_\_\_\_\_\_\_\_\_\_\ \)
(Type an exact answer, using radicals as needed.)
---
#### Explanation
To begin solving this problem, we need to express the vector-valued function \(r(t)\) in its component form and derive the parametric equations for \(x\), \(y\), and \(z\). The vector-valued function \(r(t)\) is given in terms of \(i\), \(j\), and \(k\), which represent the unit vectors along the coordinate axes.
1. **Vector Representation**:
\[
r(t) = (5i + 3j + k)
+ \cos(t) \left( \frac{1}{\sqrt{2}}i - \frac{1}{\sqrt{2}}j \right)
+ \sin(t) \left( \frac{1}{\sqrt{3}}i + \frac{1}{\sqrt{3}}j + \frac{1}{\sqrt{3}}k \right)
\]
2. **Component Form**:
\[
r(t) = 5 + \cos(t) \left( \frac{1}{\sqrt{2}} \right) + \sin(t) \left( \frac{1}{\sqrt{3}} \right)i
+ 3
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