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Lines in 3D - Concept

Teacher/Instructor Norm Prokup
Norm Prokup

Cornell University
PhD. in Mathematics

Norm was 4th at the 2004 USA Weightlifting Nationals! He still trains and competes occasionally, despite his busy schedule.

In the 3D coordinate system, lines can be described using vector equations or parametric equations. Lines in 3D have equations similar to lines in 2D, and can be found given two points on the line. In order to understand lines in 3D, one should understand how to parameterize a line in 2D and write the vector equation of a line. Perpendicular, parallel and skew lines are important cases that arise from lines in 3D.

One of the great things about the vector equation of a line is that the same equation applies in 2 dimensions as in 3 dimensions so that's our goal today is to come up with equations for lines and 3 dimensions. 3 dimensions as a vector equation is the same as in 2D here is a picture of a line in 3D all we need to know is the position vector of 1 point on the line and a direction vector that indicates the direction of the line and then the equation for the line will be r vector r equals r0 plus some constant times v the direction vector. So all we need is one position vector and a direction vector and we can get the equation of a line.
Let's see an example, find the vector and parametric equations of the line through point a 2, 3, 1 and b 5, 4, 6 and I've drawn the line here. I have to define a point p this is just going to be a random point on the line and this will give me my r my vector r and I'm going to make this vector r0 you can actually go with either a or b but I'm going to go with point a. So point a is going to give me my r0, so remember the equations r equals r0 plus t times v. So I need to come up with the components for all of these vectors, now vector r is going to have components the same as the coordinates of this point x, y and z and r0 will have components the same as point a. So r0 is 2, 3, 1 but we need a direction vector, so what I'm going to do is I'm going to define vector ab as the direction vector so it's nice to have that second point you really need it.
And so v is going to be 5-2 3, 4-3 1, and 6-1 5 that's our direction vector, and then our equation becomes x, y, z equals and r0 was 2, 3, 1 plus t times the direction vector 3, 1, 5. This is the vector equation for our line, now what the parametric equations? Well remember just like in 2 dimensions you can find the parametric equations by isolating each component. So for example, so these are the parametric equations, x=2+3 times t, y=3+1 times t and z=1+5 times t and that's it. These are the parametric equations they come right out of the vector equation. So when you're asked to find both the vector equation and the parametric equation it's really not that much harder just to go ahead and write the parametric equations down. So that's it, if you know 2 points on a line you can have with the vector n parametric equations of the line.

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