Need help w/ Calc 3 problem - point of intersection of a Plane and Line

[phatj]

I'm studying for a test I have to take in a couple days, and I cannot figure out how to do this problem.

We have the line:
x=3+2t y=6-5t z=2+3t

and the Plane:
3x+2y-4z=1

I know they are not parallel (dot product of the Plane's normal vector and Line's || vector are not 0).

So how do I find the point of intersection?

 

[phatj]

.

 

[Juice Box]

Graph them and see where they intersect

 

[Furyline]

plug the line eqns into the plane eqn and solve for t.

 

[Muzzan]

Originally posted by: phatj
I'm studying for a test I have to take in a couple days, and I cannot figure out how to do this problem.

We have the line:
x=3+2t y=6-5t z=2+3t

and the Plane:
3x+2y-4z=1

I know they are not parallel (dot product of the Plane's normal vector and Line's || vector are not 0).

So how do I find the point of intersection?

If the point (x, y, z) lies on the line, then

x = 3 + 2t
y = 6 - 5t
z = 2 + 3t

for some real number t. But if it is to lie in the plane as well, then

3x + 2y - 4z = 1

must be satisfied, i.e.

3(3 + 2t) + 2(6 - 5t) - 4(2 + 3t) = 1.

Solve for t...

 

[phatj]

bah, of course!! thank you so much i am truly dumb.