How can I find the intersection of a line vector and a plane?

Question

Here is my vector:

$(-3,1,-4)+r(4,0,1)$

And my plane: Created from the following vectors:

$x: (3,0,1)+t(-1,1,2)$

$x: (0,2,-1)+s(2,-2,-4)$

$(3,0,1)+t(-1,1,2)+n(2,-2,-4)$

(Cartesian: $x+y-3=0$)

I've tried to do the following:

vector: $(-3+4r, 1, -4+r)$

plane: $(3-t+2n, t-2n, 1+2t-4n)$

e.g.:

1) $3-t+2n = -3+4r$

2) $t-2n = 1$

3) $1+2t-4n = -4+r$

But I didn't find the right solution...
The intersection point should be: $(1,1,-3)$(written in the math book)

What did I do wrong? Thanks in advance.

Answer 1

A rather straightforward way to solve this is following these 2 steps:

1) find the cartesian equation of the plane, i.e. its equation of the form $ax+by+cz+d=0$. In order to do this, just remember that the vector $(a,b,c)$ is perpendicular to the plane and observe that you can easily obtain two independent vectors $\vec v$ and $\vec w$ parallel to the plane (so that for instance $(a,b,c)=\vec v\wedge\vec w$).

2) impose that the generic point of the line satisfies the equation of the plane and solve for $t$.

I leave the details to you as an exercise.