Proving the line lies on the plane

[soulja101]

Homework Statement

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.

Homework Equations

The Attempt at a Solution

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)

 

[tiny-tim]

Hi soulja101!

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.
…
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

This is very long-winded …

you could just put u = 0 in (x, y, z) = (5, -4, 6) + u(1,4,-1), giving (5, -4, 6) immediately

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

why make it so complicated?

all you have to prove is that (5, -4, 6) minus (3, 0, 2) is a linear combination of (1,1,-1) and (2, -1, 1), and then the same for (1,4,-1)

 

[nlightner]

to see if they fit the equation of the plane. So, I plugged in x=5, y=-4, and z=6 into the parametric equations for the plane (x, y, z) = (3, 0, 2) + s(1,1,-1) + t(2, -1, 1).

After simplifying, I got:
5=3+s+2t
-4=0+s-t
6=2-s+t

Solving this system of equations, I got s=2 and t=0. Substituting these values back into the parametric equation for the plane, I got:
(x, y, z) = (3, 0, 2) + 2(1,1,-1) + 0(2, -1, 1)
(x, y, z) = (5, -4, 6)

Since the point (5,-4,6) satisfies the equation of the plane, we can conclude that the line with equation (x, y, z) = (5, -4, 6) + u(1,4,-1) does lie on the plane with equation (x, y, z) = (3, 0, 2) + s(1,1,-1) + t(2, -1, 1). This is because the parametric equation of the line is a subset of the parametric equation of the plane, meaning that all points on the line will also satisfy the equation of the plane.