Thread: 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.

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

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?

Originally Posted by JohnBlaze

Would I just pick any two random points?

Yes! I said any two points.

Originally Posted by Plato

Yes! I said any two points.

Would (1,2,3) and (4,5,6) work?

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.

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.

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?

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?