How to find where a line intersects a plane?
Understand the Problem
The question is asking for the method to determine the point where a line crosses or meets a plane in threedimensional space. This involves using equations of the line and the plane to find the intersection point mathematically.
Answer
The intersection point is given by: $$ \mathbf{P} = \begin{pmatrix} x_0 \\ y_0 \\ z_0 \end{pmatrix} + \left(\frac{A x_0 + B y_0 + C z_0 + D}{A a + B b + C c}\right) \begin{pmatrix} a \\ b \\ c \end{pmatrix} $$
Answer for screen readers
The intersection point of the line and the plane can be represented as:
$$ \mathbf{P} = \begin{pmatrix} x_0 \ y_0 \ z_0 \end{pmatrix} + \left(\frac{A x_0 + B y_0 + C z_0 + D}{A a + B b + C c}\right) \begin{pmatrix} a \ b \ c \end{pmatrix} $$
Steps to Solve

Identify the equations of the line and the plane
Let the equation of the line be represented parametrically as: $$ \mathbf{L}(t) = \begin{pmatrix} x_0 \ y_0 \ z_0 \end{pmatrix} + t \begin{pmatrix} a \ b \ c \end{pmatrix} $$
where $(x_0, y_0, z_0)$ is a point on the line and $\begin{pmatrix} a \ b \ c \end{pmatrix}$ is the direction vector of the line.
The equation of the plane can be written as:
$$ Ax + By + Cz + D = 0 $$
where $A$, $B$, and $C$ are the coefficients of the plane's normal vector and $D$ is a constant. 
Substitute the line's equation into the plane's equation
Substitute the parametric equations from the line into the plane's equation. This involves figuring out what $x$, $y$, and $z$ represent in terms of the parameter $t$.
For example:
$$ A(x_0 + at) + B(y_0 + bt) + C(z_0 + ct) + D = 0 $$ 
Solve for the parameter $t$
Rearranging the equation gives: $$ A x_0 + B y_0 + C z_0 + D + (A a + B b + C c)t = 0 $$
Carrying out the algebra to isolate $t$, we get: $$ t = \frac{A x_0 + B y_0 + C z_0 + D}{A a + B b + C c} $$ 
Find the intersection point
Use the value of $t$ you found in the previous step and substitute it back into the parametric equation of the line to find the coordinates of the intersection point: $$ \mathbf{P} = \begin{pmatrix} x_0 \ y_0 \ z_0 \end{pmatrix} + t \begin{pmatrix} a \ b \ c \end{pmatrix} $$ 
Evaluate the coordinates
This will give you the coordinates at which the line intersects the plane.
The intersection point of the line and the plane can be represented as:
$$ \mathbf{P} = \begin{pmatrix} x_0 \ y_0 \ z_0 \end{pmatrix} + \left(\frac{A x_0 + B y_0 + C z_0 + D}{A a + B b + C c}\right) \begin{pmatrix} a \ b \ c \end{pmatrix} $$
More Information
Finding the intersection between a line and a plane in threedimensional space is crucial in various fields such as physics, engineering, and computer graphics. The method described is commonly used in 3D modeling and simulations.
Tips
 Forgetting to substitute correctly: Ensure that each variable from the line is correctly substituted into the plane's equation.
 Not checking for parallelism: If the denominator $A a + B b + C c$ equals zero, the line is either parallel to the plane or lies in it. This should be checked before finding $t$.