[C] Consider the line with parametric equations: x = 3t + 1, y = 2t −5, z = 3 + t. (C.1) Show that the point (7,-1,5) is on this line. Include a one sentence explanation of how your work demonstrates the point is on the line. (C.2) Find the coordinates of one of the points on this line that is 1 unit away from (7,−1, 5). Round the coordinates of your answer to the nearest hundredth. [Hint: a certain unit vector or it's negative might help.] (C.3) Find the equation of the plane that passes through the point (2, 7, 1) and that is perpendicular to the line whose parametric equations are given above.
[C] Consider the line with parametric equations: x = 3t + 1, y = 2t −5, z = 3 + t. (C.1) Show that the point (7,-1,5) is on this line. Include a one sentence explanation of how your work demonstrates the point is on the line. (C.2) Find the coordinates of one of the points on this line that is 1 unit away from (7,−1, 5). Round the coordinates of your answer to the nearest hundredth. [Hint: a certain unit vector or it's negative might help.] (C.3) Find the equation of the plane that passes through the point (2, 7, 1) and that is perpendicular to the line whose parametric equations are given above.
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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![## Line with Parametric Equations
Consider the line with parametric equations:
\[
x = 3t + 1, \quad y = 2t - 5, \quad z = 3 + t.
\]
### Tasks
**(C.1)** Show that the point \((7, -1, 5)\) is on this line. Include a one-sentence explanation of how your work demonstrates the point is on the line.
**(C.2)** Find the coordinates of one of the points on this line that is 1 unit away from \((7, -1, 5)\). Round the coordinates of your answer to the nearest hundredth.
[Hint: a certain unit vector or its negative might help.]
**(C.3)** Find the equation of the plane that passes through the point \((2, 7, 1)\) and that is perpendicular to the line whose parametric equations are given above.
---
### Detailed Explanation of Graphs or Diagrams
**(C.1)** To determine whether the point \((7, -1, 5)\) is on the line, substitute \(x\), \(y\), and \(z\) into their respective parametric equations and solve for \(t\). If a consistent value of \(t\) is found for all equations, then the point lies on the line.
**(C.2)** To find a point that is 1 unit away from \((7, -1, 5)\), use the concept of unit vectors. Determine a direction vector on the line, normalize it to create a unit vector, and then compute the point 1 unit away in the direction of this vector (or its negative).
**(C.3)** For the equation of the plane, use the point \((2, 7, 1)\) and the direction vector of the line. The plane equation will be derived using the vector normal to both the plane and given line, established via the direction vector.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F093e7116-3e01-49ad-9157-7a22a5caeb16%2F833f838d-0bc4-42ae-8cc4-1724fa3bf05d%2Fmj1us4k_processed.png&w=3840&q=75)
Transcribed Image Text:## Line with Parametric Equations
Consider the line with parametric equations:
\[
x = 3t + 1, \quad y = 2t - 5, \quad z = 3 + t.
\]
### Tasks
**(C.1)** Show that the point \((7, -1, 5)\) is on this line. Include a one-sentence explanation of how your work demonstrates the point is on the line.
**(C.2)** Find the coordinates of one of the points on this line that is 1 unit away from \((7, -1, 5)\). Round the coordinates of your answer to the nearest hundredth.
[Hint: a certain unit vector or its negative might help.]
**(C.3)** Find the equation of the plane that passes through the point \((2, 7, 1)\) and that is perpendicular to the line whose parametric equations are given above.
---
### Detailed Explanation of Graphs or Diagrams
**(C.1)** To determine whether the point \((7, -1, 5)\) is on the line, substitute \(x\), \(y\), and \(z\) into their respective parametric equations and solve for \(t\). If a consistent value of \(t\) is found for all equations, then the point lies on the line.
**(C.2)** To find a point that is 1 unit away from \((7, -1, 5)\), use the concept of unit vectors. Determine a direction vector on the line, normalize it to create a unit vector, and then compute the point 1 unit away in the direction of this vector (or its negative).
**(C.3)** For the equation of the plane, use the point \((2, 7, 1)\) and the direction vector of the line. The plane equation will be derived using the vector normal to both the plane and given line, established via the direction vector.
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