# Parametric Equation of a line, Conditions

• Destroxia
In summary, the parametric equations for the line through P1 and P2 are x = -4t, y = 2 - 6t for point (0,2) and x = 7, y = -3, z = 9 - 8t for point (7,-3,9). The conditions for the parametric equations for the line segment joining those points are 0 ≤ t ≤ 1. These conditions hold true for any linear algebra problem.
Destroxia

## Homework Statement

Find parametric equations for the line through P1 and P2 and also for the line segment joining those points. Also find the conditions for the parametric equations.

a) P1(0, 2) and P2(-4,-4)

b) P1(7,-3,9) and P2(7, -3, 1)

## Homework Equations

Equation of a line (Vector Form): ## \vec r = \vec r_0 + t\vec v ##[/B]

## The Attempt at a Solution

[/B]
I've already solved the problem, my issue just lies in finding the conditions for the parametric equations as well.

a) x = -4t, y = 2 - 6t

b) x = 7, y = -3, z = 9 - 8t

Wouldn't the parametric equation work from ## -\infty ≤ t ≤ \infty ##

I see no reason why some parallel direction vector wouldn't be able to be multiplied by some scalar to equal the line vector. Any multiple, negative or positive should be able to give me any line that passes through those points.

So am I correct in my assumption that the conditions should be from ## -\infty ≤ t ≤ \infty ##

RyanTAsher said:

## Homework Statement

Find parametric equations for the line through P1 and P2 and also for the line segment joining those points. Also find the conditions for the parametric equations.

a) P1(0, 2) and P2(-4,-4)

b) P1(7,-3,9) and P2(7, -3, 1)

## Homework Equations

Equation of a line (Vector Form): ## \vec r = \vec r_0 + t\vec v ##[/B]

## The Attempt at a Solution

[/B]
I've already solved the problem, my issue just lies in finding the conditions for the parametric equations as well.

a) x = -4t, y = 2 - 6t

b) x = 7, y = -3, z = 9 - 8t

Wouldn't the parametric equation work from ## -\infty ≤ t ≤ \infty ##

I see no reason why some parallel direction vector wouldn't be able to be multiplied by some scalar to equal the line vector. Any multiple, negative or positive should be able to give me any line that passes through those points.

So am I correct in my assumption that the conditions should be from ## -\infty ≤ t ≤ \infty ##
For the lines through those two sets of points, yes, but the conditions they're asking about are for the line segments. For the first problem, if t = 0, you get point P2. What value of t (the parameter) gets you P1? All of the t values in between get you the rest of the points on that line segment.

Mark44 said:
For the lines through those two sets of points, yes, but the conditions they're asking about are for the line segments. For the first problem, if t = 0, you get point P2. What value of t (the parameter) gets you P1? All of the t values in between get you the rest of the points on that line segment.

I think I see what you mean... so we are just solving for t at each end of the line segment?

it is [0,1] for all t, any linear algebra book will have this fact in the first section.

## 1. What is a parametric equation of a line?

A parametric equation of a line is a mathematical representation of a straight line in a two-dimensional plane. It uses two parameters, typically represented as t and s, to express the coordinates of any point on the line.

## 2. How is a parametric equation of a line different from a standard equation of a line?

In a standard equation of a line, the coordinates of any point on the line are expressed in terms of x and y. In a parametric equation, the coordinates are expressed in terms of two parameters, t and s.

## 3. What are the conditions for a parametric equation of a line to be valid?

In order for a parametric equation of a line to be valid, the two parameters t and s must be independent, meaning that changes in one do not affect the other. Additionally, the equation must represent a straight line, so the parameters must be linearly related.

## 4. How do you graph a parametric equation of a line?

To graph a parametric equation of a line, you can plot points by plugging in different values for the parameters and then connecting them with a straight line. Alternatively, you can use a graphing calculator or software to graph the equation.

## 5. Can parametric equations of a line be used in three-dimensional space?

Yes, parametric equations of a line can be used in three-dimensional space. In this case, the equation would use three parameters, typically represented as t, s, and r, to express the coordinates of any point on the line in three-dimensional space.

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