Consider the line which pases through the Parallel to the line X = 1 +7+₁₂y= 2 + + z=3+7+ find the of of this the Xy plane ху • XZ plane A point of intersection which each coordinate planes new line Iz plane ₁0
Consider the line which pases through the Parallel to the line X = 1 +7+₁₂y= 2 + + z=3+7+ find the of of this the Xy plane ху • XZ plane A point of intersection which each coordinate planes new line Iz plane ₁0
Advanced Engineering Mathematics
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Author:Erwin Kreyszig
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Chapter2: Second-order Linear Odes
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![**Title: Finding the Intersection Points of a Line with Coordinate Planes**
**Problem Statement:**
Consider the line which passes through the point (5, 5, -1) and which is parallel to the line given by the parametric equations:
\[ X = 1 + 7t \]
\[ Z = 3 + t \]
\[ 2t + Z = 3t + 7 \]
**Objective:**
Find the point of intersection where this new line intersects each of the coordinate planes:
1. **XZ-plane**
2. **XY-plane**
3. **YZ-plane**
**Solution Steps:**
To find the points of intersection, we need to set the corresponding coordinate related to the plane being zero and solve for the other variables.
For example:
- **XZ-plane**: Set `y = 0` and solve for `x` and `z`.
- **XY-plane**: Set `z = 0` and solve for `x` and `y`.
- **YZ-plane**: Set `x = 0` and solve for `y` and `z`.
The intersection points are then calculated by substituting these values into the parametric equations given for the line.
**Graphical Representation:**
A 3D coordinate system might typically represent these solution points, showcasing where the line pierces through each respective coordinate plane. It visually helps assess the spatial orientation and intersection of the line with each plane.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F99d15f92-0bff-4b4d-a47e-2ac33d144271%2F1fe4f401-cbb8-42ac-9c49-425d337456ed%2F9ocndpo_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Title: Finding the Intersection Points of a Line with Coordinate Planes**
**Problem Statement:**
Consider the line which passes through the point (5, 5, -1) and which is parallel to the line given by the parametric equations:
\[ X = 1 + 7t \]
\[ Z = 3 + t \]
\[ 2t + Z = 3t + 7 \]
**Objective:**
Find the point of intersection where this new line intersects each of the coordinate planes:
1. **XZ-plane**
2. **XY-plane**
3. **YZ-plane**
**Solution Steps:**
To find the points of intersection, we need to set the corresponding coordinate related to the plane being zero and solve for the other variables.
For example:
- **XZ-plane**: Set `y = 0` and solve for `x` and `z`.
- **XY-plane**: Set `z = 0` and solve for `x` and `y`.
- **YZ-plane**: Set `x = 0` and solve for `y` and `z`.
The intersection points are then calculated by substituting these values into the parametric equations given for the line.
**Graphical Representation:**
A 3D coordinate system might typically represent these solution points, showcasing where the line pierces through each respective coordinate plane. It visually helps assess the spatial orientation and intersection of the line with each plane.
![### Problem Statement
**Task:**
Consider the line which passes through the point \( P(5, -1, 1) \) and which is parallel to the line:
\[ x = 1 + 7t, \]
\[ y = 2 + t, \]
\[ z = 3 + 2t. \]
Find the point of intersection of this new line with each of the coordinate planes.
**Coordinate Planes:**
- **XY Plane:** \( z = 0 \)
- **XZ Plane:** \( y = 0 \)
- **YZ Plane:** \( x = 0 \)
**Guidance:**
For each plane, substitute the appropriate value (0 for the variable not included in the plane) into the line equations to find the points of intersection. Fill the following boxes with your answers:
**Results:**
- XY plane: \([ \ \ ]\)
- XZ plane: \([ \ \ ]\)
- YZ plane: \([ \ \ ]\)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F99d15f92-0bff-4b4d-a47e-2ac33d144271%2F1fe4f401-cbb8-42ac-9c49-425d337456ed%2Fq21u4lf_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Problem Statement
**Task:**
Consider the line which passes through the point \( P(5, -1, 1) \) and which is parallel to the line:
\[ x = 1 + 7t, \]
\[ y = 2 + t, \]
\[ z = 3 + 2t. \]
Find the point of intersection of this new line with each of the coordinate planes.
**Coordinate Planes:**
- **XY Plane:** \( z = 0 \)
- **XZ Plane:** \( y = 0 \)
- **YZ Plane:** \( x = 0 \)
**Guidance:**
For each plane, substitute the appropriate value (0 for the variable not included in the plane) into the line equations to find the points of intersection. Fill the following boxes with your answers:
**Results:**
- XY plane: \([ \ \ ]\)
- XZ plane: \([ \ \ ]\)
- YZ plane: \([ \ \ ]\)
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