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In Exercises 21 and $22,$ find a parametric equation of the line $M$ through $\mathbf{p}$ and $\mathbf{q} \cdot[\text { Hint: } M \text { is parallel to the vector } \mathbf{q}-\mathbf{p} . \text { See the }$ figure below. $$ \mathbf{p}=\left[\begin{array}{r}{-6} \\ {3}\end{array}\right], \mathbf{q}=\left[\begin{array}{r}{0} \\ {-4}\end{array}\right] $$

Name: Problem 22, Chapter 1: Linear Algebra and Its Applications
Uploaded: 2023-09-01T23:11:12-08:00
Duration: 1 min 5 s
Channel: William Mead
Description: In Exercises 21 and 22, find a parametric equation of the line M through 𝐩 and 𝐪 ·[ Hint: M is parallel to the vector 𝐪-𝐩 . See the figure below. 𝐩=[ -6 3 ], 𝐪=[ 0 -4 ]

   In Exercises 21 and $22,$ find a parametric equation of the line $M$ through $\mathbf{p}$ and $\mathbf{q} \cdot[\text { Hint: } M \text { is parallel to the vector } \mathbf{q}-\mathbf{p} . \text { See the }$ figure below. 
$$
\mathbf{p}=\left[\begin{array}{r}{-6} \\ {3}\end{array}\right], \mathbf{q}=\left[\begin{array}{r}{0} \\ {-4}\end{array}\right]
$$

Linear Algebra and Its Applications

David C. Lay, Steven… 5th Edition

Chapter 1, Problem 22

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Solved by Expert William Mead

09/01/2023

Step 1

This vector is parallel to the line $M$ we are looking for. We can find this vector by subtracting the coordinates of $\mathbf{p}$ from the coordinates of $\mathbf{q}$: $$ \mathbf{q}-\mathbf{p}=\left[\begin{array}{r}{0} \\ Show more…

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In Exercises 21 and $22,$ find a parametric equation of the line $M$ through $\mathbf{p}$ and $\mathbf{q} \cdot[\text { Hint: } M \text { is parallel to the vector } \mathbf{q}-\mathbf{p} . \text { See the }$ figure below. $$ \mathbf{p}=\left[\begin{array}{r}{-6} \\ {3}\end{array}\right], \mathbf{q}=\left[\begin{array}{r}{0} \\ {-4}\end{array}\right] $$

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Key Concepts

Parametric Representation of a Line

This concept involves expressing a line in terms of a point and a direction vector. The equation is typically written as r(t) = r0 + t*v, where r0 is a point on the line, v is the direction vector, and t is a real number. This representation allows one to generate every point on the line by varying the parameter t.

Direction Vector

The direction vector is a vector that indicates the direction in which the line extends. It is often found by subtracting the coordinates of one point from another on the line. This vector is crucial for constructing the parametric equation since it defines the line's orientation in space.

Vector Operations

Key operations such as vector subtraction and scalar multiplication are used to compute the direction vector and to scale it appropriately in the parametric equation. These operations are fundamental in linear algebra and are essential for manipulating vectors in coordinate geometry.

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