Find a parametrization r(t) of the line through the origin whose projection on the xy-plane is a line of slope 9 and on the yz-plane is a line of slope 4 (i.e. = 4). Consider the first component of r(t) is t. (Use symbolic notation and fractions where needed.) The 2nd component of r(t) = The 3rd component of r(t) =
Find a parametrization r(t) of the line through the origin whose projection on the xy-plane is a line of slope 9 and on the yz-plane is a line of slope 4 (i.e. = 4). Consider the first component of r(t) is t. (Use symbolic notation and fractions where needed.) The 2nd component of r(t) = The 3rd component of r(t) =
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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![**Parametrization of a Line through the Origin**
To find a parametrization \( \mathbf{r}(t) \) of the line through the origin, whose projection on the \( xy \)-plane has a slope of 9 and on the \( yz \)-plane has a slope of 4 (\( \frac{\Delta z}{\Delta y} = 4 \)), follow the steps below. Considering the first component of \( \mathbf{r}(t) \) is \( t \):
1. **Understanding the Slopes Given:**
- The slope of the projection on the \( xy \)-plane (\( m_{xy} \)) is given as 9.
- The slope of the projection on the \( yz \)-plane (\( m_{yz} \)) is given as 4 \( \left( \frac{\Delta z}{\Delta y} = 4 \right) \).
2. **Determining the Components of \( \mathbf{r}(t) \):**
- Since the first component (x-coordinate) of the line is \( t \), use the slopes to determine the second (y-coordinate) and third (z-coordinate) components.
3. **Second Component (y-coordinate):** Based on the slope \( m_{xy} = 9 \), for an increase in \( x \) by \( t \), the change in \( y \) would be:
\[
y = 9t
\]
4. **Third Component (z-coordinate):** Based on the slope \( m_{yz} = 4 \), for an increase in \( y \) by \( 9t \), the change in \( z \) would be:
\[
z = 4(9t) = 36t
\]
5. **Expressing \( \mathbf{r}(t) \):**
\[
\mathbf{r}(t) = (t, 9t, 36t)
\]
**Determine the components:**
- The **2nd component** of \( \mathbf{r}(t) \) = \( 9t \)
- The **3rd component** of \( \mathbf{r}(t) \) = \( 36t \)
These expressions can now be used to parametrize the given line in a 3-dimensional space.
**Discussion Section:**
Understanding how](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F657f96d0-e702-4515-9e22-2987647efb37%2Fe8b657f1-1246-4496-afda-9f7bbb3464ae%2Fbcmhwkj_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Parametrization of a Line through the Origin**
To find a parametrization \( \mathbf{r}(t) \) of the line through the origin, whose projection on the \( xy \)-plane has a slope of 9 and on the \( yz \)-plane has a slope of 4 (\( \frac{\Delta z}{\Delta y} = 4 \)), follow the steps below. Considering the first component of \( \mathbf{r}(t) \) is \( t \):
1. **Understanding the Slopes Given:**
- The slope of the projection on the \( xy \)-plane (\( m_{xy} \)) is given as 9.
- The slope of the projection on the \( yz \)-plane (\( m_{yz} \)) is given as 4 \( \left( \frac{\Delta z}{\Delta y} = 4 \right) \).
2. **Determining the Components of \( \mathbf{r}(t) \):**
- Since the first component (x-coordinate) of the line is \( t \), use the slopes to determine the second (y-coordinate) and third (z-coordinate) components.
3. **Second Component (y-coordinate):** Based on the slope \( m_{xy} = 9 \), for an increase in \( x \) by \( t \), the change in \( y \) would be:
\[
y = 9t
\]
4. **Third Component (z-coordinate):** Based on the slope \( m_{yz} = 4 \), for an increase in \( y \) by \( 9t \), the change in \( z \) would be:
\[
z = 4(9t) = 36t
\]
5. **Expressing \( \mathbf{r}(t) \):**
\[
\mathbf{r}(t) = (t, 9t, 36t)
\]
**Determine the components:**
- The **2nd component** of \( \mathbf{r}(t) \) = \( 9t \)
- The **3rd component** of \( \mathbf{r}(t) \) = \( 36t \)
These expressions can now be used to parametrize the given line in a 3-dimensional space.
**Discussion Section:**
Understanding how
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