For questions involving lines and planes, we are usually asked to find the point of intersection. However, when we consider a line and a plane, there are three possible situations. In this page, we will consider all the possible outcomes.
1) the line intersects the plane
2) the line is parallel to the plane
3) the line lies in the plane
Key Concepts
On this page, you should learn about
the intersection of a line and a plane
Essentials
The following videos will help you understand all the concepts from this page
A Line and a Plane
In the following video, we consider the three different possibilities
Example 1 - Point of Intersection
Find the point of intersection of the line \(-x=\frac {y-5}{2}=2z-8\) with the plane 3x – y + z = 8
Example 2 - No Intersection
Find the point of intersection of the line \(\textbf{r}=\left( \begin{matrix} 2 \\ 1\\0 \end{matrix} \right) +\mu\left( \begin{matrix} 4 \\ 3 \\1\end{matrix} \right) \)with the plane x - 2y + 2z = 6
Example 3 - Line lies in Plane
Find the point of intersection of the line \(x=2+4λ\ ,\ y=1+3λ\ ,\ z=λ\) with the plane x - 2y + 2z = 0
In the following video, we look at a specific application of finding the intersection of a line with a plane. We will see how it is possible to find the coordinates of the point reflected in a plane. In order to do this, we have to find the equation of a line that goes through our point and is perpendicular to the plane.
Here is the example:
The point A (1, 3, 0) is on the line L, which is perpendicular to the plane \(3x−3y+2z=5\).
Find the equation of the line L.
Find the point R which is the intersection of the line L and the plane.
The point A is reflected in the plane. Find the coordinates of the image of A.
Find the value of \(\lambda\) in order that the point (6,1,-1) lies on the line \(\textbf{r}=\left( \begin{matrix} 2\\3 \\ -1 \end{matrix} \right) +\lambda \left( \begin{matrix} -2 \\ 1\\0 \end{matrix} \right) \)
The point (2 , a , b ) lies on the line \(\textbf{r}=\left( \begin{matrix} -1\\0 \\ 2 \end{matrix} \right) +\lambda \left( \begin{matrix} 1 \\ -1\\2 \end{matrix} \right) \)
(0 , a , b) is the point where the line line \(\textbf{r}=\left( \begin{matrix} 2\\1 \\ 4 \end{matrix} \right) +\lambda\left( \begin{matrix} -1 \\ 2\\3 \end{matrix} \right) \) meets the plane x = 0.