# Proving this line lies on this plane with a specfic equaction

• Mar 29th 2009, 11:16 AM
JohnBlaze
Proving this line lies on this plane with a specfic equaction
Q.Does the line with equation (x, y, z) = (5, -4, 6) + u(1,4,-1) lie in the plane with equation (x, y, z) = (3, 0, 2) + s(1,1,-1) + t(2, -1, 1)? Justify your answer algebraically.

What I tried doing:
I started by getting the parametric equation of (x, y, z) = (5, -4, 6) + u(1,4,-1)
x=5+u
y=-4+4u
z=6-u
I then subbed in u=0 to get a set of points
x=5
y=-4
z=6

I then got the parametric equation for (x, y, z) = (3, 0, 2) + s(1,1,-1) + t(2, -1, 1)
x=3+s+2t
y=0+s-t
z=2-s+t

I decided to use the points (5,-4,6) and sub them into
x=3+s+2t
y=0+s-t
z=2-s+t

5=3+s+2t
5-3=s+2t
2=s+2t

-4=0+s-t
-4+0=s-t
-4=s-t
6=2-s+t
6-2=-s+t
4=-s+t

I don't know if I'm doing this correctly. I really don't know what to so I tried that.(Thinking)
• Mar 29th 2009, 11:22 AM
Plato
All you have to do is take any two points on the line and show those points are on the plane.
• Mar 29th 2009, 11:24 AM
JohnBlaze
Quote:

Originally Posted by Plato
All you have to do is take any two points on the line and show those points are on the plane.

How would I do that? Would I just pick any two random points?
• Mar 29th 2009, 11:44 AM
Plato
Quote:

Originally Posted by JohnBlaze
Would I just pick any two random points?

Yes! I said any two points.
• Mar 29th 2009, 12:01 PM
JohnBlaze
Quote:

Originally Posted by Plato
Yes! I said any two points.

Would (1,2,3) and (4,5,6) work?
• Mar 29th 2009, 12:05 PM
Plato
Quote:

Originally Posted by JohnBlaze
Would (1,2,3) and (4,5,6) work?

Do they belong on the line?
The points must of course be on the line..
Theorem: If two points of a line are on a plane then the line is a subset of the plane.
• Mar 29th 2009, 12:09 PM
HallsofIvy
Quote:

Originally Posted by JohnBlaze
How would I do that? Would I just pick any two random points?

Choose two value for u and calculate the coordinates of the points from your parametric equations. If both points satisfy the equation of the plane, then the entire line is in the plane.
• Mar 29th 2009, 12:19 PM
JohnBlaze
Quote:

Originally Posted by HallsofIvy
Choose two value for u and calculate the coordinates of the points from your parametric equations. If both points satisfy the equation of the plane, then the entire line is in the plane.

I chose two values for u(0 and 1)
the sets of coordinates I got are (5,-4,6) and (6,4, 5) Is that right? How would I verify that the points satisfy the equation of the plane?
• Mar 29th 2009, 12:36 PM
JohnBlaze
Quote:

Originally Posted by Plato
Do they belong on the line?
The points must of course be on the line..
Theorem: If two points of a line are on a plane then the line is a subset of the plane.

So I should pick values for u and then solve the equation?