Two questions involving lines and planes

[TeenieBopper]

I have two questions involving lines and planes. They're both fairly simple, but I'm stuck. I'm sure someone is going to point something out and it's going to make me smack my forehead.

Homework Statement

Where does the line through*(1, 0, 1) and (4,*−2,*4) intersect the plane*x*+*y*+*z*=*10?

Homework Equations

The Attempt at a Solution

Okay, I know I need to get the equation of the line between (1,0,1) and (4, -2, 4). I find the direction vector to be <3, -2, 3>. Now, r(t)=r0+tv, which, using (1,0,1) as r), I find to be:

r(t)=(1-3t, 2t, 1-3t)

Once I have those, I simply plug those values into the equation of the plane to find t.

(1-3t)+(-2t)+1-3t)=10
2-8t=10
t=-1

And now I take that value of t and plug it into t(t) to get (x,y,z) coordinates. So (x,y,z)=(4, -2, 4).

However, (4, -2, 4) is incorrect.

Homework Statement

Consider the following planes.
4x*−*3y*+*z*=*1 and*****3x*+*y*−*4z*=*4
(a) Find parametric equations for the line of intersection of the planes. (Use the parameter*t.)

Homework Equations

The Attempt at a Solution

So n1= <4, -3, 1> and n2=<3, 1, -4>, and I can find the direction of the line of intersection by finding the cross product of n1Xn2, which is <-13, 19, -5>. However, to find the parametric equation of the line, I still need r0, but I don't know how to find a value that is on the line.Thanks in advance for the help.

 

[Muphrid]

For the first problem, you put -2t instead of 2t when you substituted into the equation of the plane to try to solve for t.

For the second problem, one thing that springs to mind is you can set z=0. Each of the planes cuts that plane in a line, and those two lines have a unique point of intersection, which is where the common line between the planes cuts the z=0 plane.

I suspect this is not what you're intended to do, though, because it requires a little creativity. There's probably a simpler method.