Calculus: Early Transcendentals
8th Edition
ISBN: 9781285741550
Author: James Stewart
Publisher: Cengage Learning
expand_more
expand_more
format_list_bulleted
Related questions
Question
![---
### How to Parametrize a Line Using Vector Parametrization
Recall how to parametrize the line through \( P = (a_1, b_1, c_1) \) and \( Q = (a_2, b_2, c_2) \). Using vector parametrization with parameter \( t \), we have the following description of the line:
\[ \mathbf{r}(t) = (1 - t) \overrightarrow{OP} + t \overrightarrow{OQ} = (1 - t)(a_1, b_1, c_1) + t(a_2, b_2, c_2) \]
#### Example: Parametrizing the Line Through \( P = (2, 1, -1) \) and \( Q = (14, 5, 7) \)
\[ \mathbf{r}(t) = \boxed{(1 - t)(2, 1, -1)} + \boxed{t(14, 5, 7)} \]
Expanding, we get:
\[ \mathbf{r}(t) = \langle 2 + 12t, \boxed{1 + 4t, -1 + 8t} \rangle \]
---
Explanation of Equations:
1. The first equation represents the general form of a line's vector parametrization: it combines the position vector of \( P \) scaled by \( 1 - t \) and the position vector of \( Q \) scaled by \( t \).
2. The second equation provides an example using specific points \( P \) and \( Q \). The vector form illustrates the transition from the initial point \( P \) to the terminal point \( Q \).
3. Finally, the expanded equation demonstrates the step-by-step algebra involved in combining and simplifying terms for each component of the vector.
Graphical or Diagram Explanation:
- Since the image contains text equations and boxed expressions rather than graphical diagrams, the detailed description includes algebraic manipulation of the vector equation to illustrate the linear dependence on the parameter \( t \).
---](https://content.bartleby.com/qna-images/question/c3d3bee5-0c39-4b7b-a553-642953afa184/d37ac7ef-3c42-43fd-b328-09c3c40a6ebc/w0qlfmf.png)
Transcribed Image Text:---
### How to Parametrize a Line Using Vector Parametrization
Recall how to parametrize the line through \( P = (a_1, b_1, c_1) \) and \( Q = (a_2, b_2, c_2) \). Using vector parametrization with parameter \( t \), we have the following description of the line:
\[ \mathbf{r}(t) = (1 - t) \overrightarrow{OP} + t \overrightarrow{OQ} = (1 - t)(a_1, b_1, c_1) + t(a_2, b_2, c_2) \]
#### Example: Parametrizing the Line Through \( P = (2, 1, -1) \) and \( Q = (14, 5, 7) \)
\[ \mathbf{r}(t) = \boxed{(1 - t)(2, 1, -1)} + \boxed{t(14, 5, 7)} \]
Expanding, we get:
\[ \mathbf{r}(t) = \langle 2 + 12t, \boxed{1 + 4t, -1 + 8t} \rangle \]
---
Explanation of Equations:
1. The first equation represents the general form of a line's vector parametrization: it combines the position vector of \( P \) scaled by \( 1 - t \) and the position vector of \( Q \) scaled by \( t \).
2. The second equation provides an example using specific points \( P \) and \( Q \). The vector form illustrates the transition from the initial point \( P \) to the terminal point \( Q \).
3. Finally, the expanded equation demonstrates the step-by-step algebra involved in combining and simplifying terms for each component of the vector.
Graphical or Diagram Explanation:
- Since the image contains text equations and boxed expressions rather than graphical diagrams, the detailed description includes algebraic manipulation of the vector equation to illustrate the linear dependence on the parameter \( t \).
---
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution
Trending nowThis is a popular solution!
Step by stepSolved in 2 steps with 2 images
Knowledge Booster
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, calculus and related others by exploring similar questions and additional content below.Similar questions
- please show all your workvectorsarrow_forwardPlease don't provide handwritten solution ....arrow_forwardPlease look at the attached image as its clearer Suppose the straight lines L1�1 and L2�2 have respective vector equations r1=⎛⎝⎜4−33⎞⎠⎟+t⎛⎝⎜5−11⎞⎠⎟andr2=⎛⎝⎜98−2⎞⎠⎟+s⎛⎝⎜−5−52⎞⎠⎟,�1=(4−33)+�(5−11)and�2=(98−2)+�(−5−52), where s� and t� are scalar parameters. a) Given that L1�1 and L2�2 intersect at some point P�, find its coordinates. P=�=(,,) b) Find the exact value of the cosine of the acute angle θ�, between L1�1 and L2�2. cosθ=cos�=arrow_forward
arrow_back_ios
arrow_forward_ios
Recommended textbooks for you
- Calculus: Early TranscendentalsCalculusISBN:9781285741550Author:James StewartPublisher:Cengage Learning
Thomas' Calculus (14th Edition)CalculusISBN:9780134438986Author:Joel R. Hass, Christopher E. Heil, Maurice D. WeirPublisher:PEARSON
Calculus: Early Transcendentals (3rd Edition)CalculusISBN:9780134763644Author:William L. Briggs, Lyle Cochran, Bernard Gillett, Eric SchulzPublisher:PEARSON 
Calculus: Early TranscendentalsCalculusISBN:9781319050740Author:Jon Rogawski, Colin Adams, Robert FranzosaPublisher:W. H. Freeman
Calculus: Early Transcendental FunctionsCalculusISBN:9781337552516Author:Ron Larson, Bruce H. EdwardsPublisher:Cengage Learning
Calculus: Early Transcendentals
Calculus
ISBN:9781285741550
Author:James Stewart
Publisher:Cengage Learning
Thomas' Calculus (14th Edition)
Calculus
ISBN:9780134438986
Author:Joel R. Hass, Christopher E. Heil, Maurice D. Weir
Publisher:PEARSON
Calculus: Early Transcendentals (3rd Edition)
Calculus
ISBN:9780134763644
Author:William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz
Publisher:PEARSON
Calculus: Early Transcendentals
Calculus
ISBN:9781319050740
Author:Jon Rogawski, Colin Adams, Robert Franzosa
Publisher:W. H. Freeman
Calculus: Early Transcendental Functions
Calculus
ISBN:9781337552516
Author:Ron Larson, Bruce H. Edwards
Publisher:Cengage Learning