Intersection of Line and Plane

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

Notes from the video

 

Reflecting a Point in a Plane

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 \(3x−3y+2z=5\).

  1. Find the equation of the line L.
  2. Find the point R which is the intersection of the line L and the plane.
  3. The point A is reflected in the plane. Find the coordinates of the image of A.

Notes from the video

 

Summary

Print from here

Test Yourself

Here is a quiz that practises the skills from this page


START QUIZ!

Exam-style Questions

Question 1

The points A and B are given by A(-8,1,-2) and B(-2,-1,2).

A plane Π is defined by the equation \(2x−y−3z=−8\)

  1. Find a vector equation of the line L passing through the points A and B.
  2. Find the coordinates of the point of intersection of the line and the plane.

Hint

Full Solution

Question 2

Consider the plane \(x−2y+4z=−15\) and the line

\(x=3+kλ\\ y=−2+λ\\ z=(2k+6)−2λ\)

The line and the plane are perpendicular. Find

  1. The value of k
  2. The coordinates of the point of intersection of the line and the plane.

Hint

Full Solution

Question 3

\(Π_{ 1 }\)and \(Π_{ 2 }\) are planes such that

\(Π_{ 1 }:2x−y−2z=0\)

and

\(Π_{ 2 }:−2x+3y+3z=4 \)

L is the intersection of planes \(Π_{ 1 }\) and \(Π_{ 2 }\)

  1. Find the equation of the line

A third plane \(Π_{ 3 }\) is defined by the equation \(kx+(k−1)y−z=5\)

  1. Find the value of k such that the line L does not intersect with \(Π_{ 3 }\)

Hint

Full Solution

Question 4

The point A (3, 1, –2) is on the line L, which is perpendicular to the plane \(2x−3y−z+9=0\).

  1. Find the Cartesian equation of the line L.
  2. Find the point R which is the intersection of the line L and the plane.
  3. The point A is reflected in the plane. Find the coordinates of the image of A.

Hint

Full Solution

MY PROGRESS

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