Finding the Equation of a 3D Line from Two Points

In summary, the conversation discusses finding the equation of a line in three dimensional space using a parametric equation. The equation c(t) = (P-Q)t+P is valid, but it can also be written as c(t) = (P-Q)t+Q. The expert also mentions that since a line is one-dimensional, in a three dimensional space, multiple equations are needed. The expert explains the use of a parameter and how it can be used to calculate the coordinates of a point on the line.
  • #1
tandoorichicken
245
0
How do you find the equation of a line between two points in three dimensional space? I sort of forgot. =\
 
Physics news on Phys.org
  • #2
c(t) = (P-Q)t+P
 
  • #3
0rthodontist said:
c(t) = (P-Q)t+P

Shouldn't that be c(t) = (P-Q)t+Q

because at t = 0 you should get Q, and at t = 1 you should get P which isn't what happens in your equation.
 
  • #4
It really doesn't matter how you parametrize it so long as it is the same line. c(t) = .3455(P-Q)t+(P+Q)/2 is equally valid.
 
  • #5
d_leet said:
Shouldn't that be c(t) = (P-Q)t+Q

because at t = 0 you should get Q, and at t = 1 you should get P which isn't what happens in your equation.

Orthodontist's form, c= (P- Q)t+ P, gives P when t=0 and Q when t= -1. That perfectly valid.

By the way, Tandoorichicken, since a line is one-dimensional, in a three dimensional space you need more than one equation. Orthodontist gave a "vector" equation where P and Q are the position vectors of two points and t is a parameter. Writing that in component form gives three parametric equations. Given the single variable t, you can calculate x, y, and z.
 
Last edited by a moderator:

FAQ: Finding the Equation of a 3D Line from Two Points

What is the equation of a 3D line?

The equation of a 3D line is a mathematical representation that describes the relationship between the x, y, and z coordinates of points that lie on that line.

How many points are needed to find the equation of a 3D line?

At least two points are needed to find the equation of a 3D line. These two points must lie on the line and have distinct x, y, and z coordinates.

Can the equation of a 3D line be found if the points are on the same plane?

No, the equation of a 3D line cannot be found if the points are on the same plane. A line on a plane can be represented by a different equation, such as a parametric equation.

What are the steps for finding the equation of a 3D line from two points?

The steps for finding the equation of a 3D line from two points are:
1. Find the slope of the line using the two points
2. Choose one point and use the slope to write the equation in point-slope form
3. Rewrite the equation in general form by distributing and rearranging terms
4. Simplify the equation and write it in standard form to get the final equation.

Can the equation of a 3D line be used to find the distance between two points on that line?

Yes, the equation of a 3D line can be used to find the distance between two points on that line. Once the equation is found, the distance formula can be used to calculate the distance between the two points.

Back
Top