---### How to Parametrize a Line Using Vector ParametrizationRecall 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 $.---

Answered: Image /qna-images/answer/d37ac7ef-3c42-43fd-b328-09c3c40a6ebc.jpg

Recall how to parametrize the line through P = (a1, b1, c1) and Q = (a2, b2, c2). Using vector parametrization with parameter t, we have the following description of the line. r(t) = (1 – t)OP + tOQ = (1 – t)(a1, b1, c1) + t(a2, b2, c2) Parametrize the line through P = (2, 1, –1) and Q = (14, 5, 7). r(t) = )(2, 1, -1) + (14, 5, 7) 2 + 12t,

Recall how to parametrize the line through P = (a1, b1, c1) and Q = (a2, b2, c2). Using vector parametrization with parameter t, we have the following description of the line. r(t) = (1 – t)OP + tOQ = (1 – t)(a1, b1, c1) + t(a2, b2, c2) Parametrize the line through P = (2, 1, –1) and Q = (14, 5, 7). r(t) = )(2, 1, -1) + (14, 5, 7) 2 + 12t,

Calculus: Early Transcendentals

8th Edition

ISBN:9781285741550

Author:James Stewart

Publisher:James Stewart

Chapter1: Functions And Models

Section: Chapter Questions

Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...

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### 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:

\[ \mathbf{r}(t) = \langle 2 + 12t, \boxed{1 + 4t, -1 + 8t} \rangle \]

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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 $.

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