# Writing vector and parametric equations for a line that....

• Specter
In summary: You can check it yourself. Try ##t=0## and ##t=1## to see if it goes through your two points.Yes, that's what LCKurtz meant.In summary, the line that goes through the two points (–3, 5, 2) and (2, 7, 1) is tangent to the y-axis at point (2, 7, 1).
Specter

## Homework Statement

[/B]
Write vector and parametric equations for the line that goes through the points P(–3, 5, 2) and Q(2, 7, 1).

## The Attempt at a Solution

First I find the direction vector for PQ.

PQ=Q-P = (2,7,1)-(-3,5,2)
=[2-(-3),7-5,1-2]
=5,2,-1

PQ= (5,2,-1)

Now I have a direction vector and two points. I think I can write the vector equation now but I am not sure which point to use, P or Q, and if it matters.

If I were to use P(-3,5,2) the vector equation would look like the following:

(x,y,z)=(-3,5,2)+t(5,2,-1)

Then the parametric equations would be

x=-3+5t
y=5+2t
z=2-t

You can check it yourself. Try ##t=0## and ##t=1## to see if it goes through your two points.

Specter
LCKurtz said:
You can check it yourself. Try ##t=0## and ##t=1## to see if it goes through your two points.

Sorry if this is a dumb question but this is my first math course in... a while. Is this what you meant?

I set t=0 and t=1 for the parametric equations and got the coordinates

t=0, (-3,5,2)
t=1, (2,7,1)

Yes, that's what LCKurtz meant.

Specter said:
I am not sure which point to use, P or Q, and if it matters.
Doesn't matter -- you can use either one.

Specter
Mark44 said:
Yes, that's what LCKurtz meant.

Doesn't matter -- you can use either one.
Thank you.

Specter said:

## Homework Statement

[/B]
Write vector and parametric equations for the line that goes through the points P(–3, 5, 2) and Q(2, 7, 1).

## The Attempt at a Solution

First I find the direction vector for PQ.

PQ=Q-P = (2,7,1)-(-3,5,2)
=[2-(-3),7-5,1-2]
=5,2,-1

PQ= (5,2,-1)

Now I have a direction vector and two points. I think I can write the vector equation now but I am not sure which point to use, P or Q, and if it matters.

If I were to use P(-3,5,2) the vector equation would look like the following:

(x,y,z)=(-3,5,2)+t(5,2,-1)

Then the parametric equations would be

x=-3+5t
y=5+2t
z=2-t
Well, I wrote this reply and failed to actually post it.
Doesn't really say anything much different than what the others said.
...

Your results look fine.

It does not matter which of the points you start with. In fact you can find some other point on this line and use that.

Using s as the parameter, start at point Q: (2, 7, 1) .

Last edited:
Specter

## 1. How do you write vector equations for a line?

To write a vector equation for a line, you need to have a starting point and a direction vector. The equation will be of the form r = r0 + tv, where r0 is the starting point and v is the direction vector.

## 2. What is a parametric equation for a line?

A parametric equation for a line is a set of equations that describe the position of a point on the line in terms of one or more parameters. For a line in three-dimensional space, the parametric equations are x = x0 + at, y = y0 + bt, and z = z0 + ct, where x0, y0, and z0 are the coordinates of a point on the line and a, b, and c are the direction numbers.

## 3. How do you convert a vector equation to a parametric equation?

To convert a vector equation to a parametric equation, you need to identify the starting point and the direction vector of the line. The starting point will be the constant term in the vector equation, while the direction vector will be the coefficient of the variable t. Once you have these values, you can use the formula r0 + tv to write the parametric equation.

## 4. What is the significance of the parameter in a parametric equation?

The parameter in a parametric equation represents the variable that determines the position of a point on the line. By varying the value of the parameter, you can trace out the entire line. It also allows for a more flexible representation of the line, as different values of the parameter can result in different points on the line.

## 5. Can you write a vector equation or parametric equation for a line in two-dimensional space?

Yes, a line in two-dimensional space can be represented by both a vector equation and a parametric equation. The vector equation will have the form r = r0 + tv, where r0 is the starting point and v is the direction vector. The parametric equations will be x = x0 + at and y = y0 + bt, where x0 and y0 are the coordinates of a point on the line and a and b are the direction numbers.

• Precalculus Mathematics Homework Help
Replies
8
Views
2K
• Precalculus Mathematics Homework Help
Replies
10
Views
2K
• Precalculus Mathematics Homework Help
Replies
6
Views
806
• Precalculus Mathematics Homework Help
Replies
1
Views
762
• Precalculus Mathematics Homework Help
Replies
3
Views
1K
• Precalculus Mathematics Homework Help
Replies
7
Views
767
• Precalculus Mathematics Homework Help
Replies
9
Views
2K
• Precalculus Mathematics Homework Help
Replies
8
Views
1K
• Precalculus Mathematics Homework Help
Replies
14
Views
417
• Precalculus Mathematics Homework Help
Replies
2
Views
2K