(a) Give an example of three planes in R^3 that have a common line of intersection. Justify your answer.
(b) Give an example of three planes in R^3 that intersect in pairs but have no common point of intersection. Justify your answer.
(c) Give an example of three planes in R^3 that intersect in a single point. Justify your answer.
Here is an excellent video, with diagrams and example set of equations.
There are actually 3 videos, they should run into each other.
https://www.youtube.com/watch?v=duFRYId7kNU
(a) An example of three planes in R^3 that have a common line of intersection is:
Plane 1: x - y + z = 0
Plane 2: 2x + y - 3z = 1
Plane 3: -3x + 2y - 4z = -2
To justify, we can find the common line of intersection by solving the system of equations formed by the planes. By solving the system, we find that the line of intersection is given by the parametric equations:
x = t
y = -t
z = 2t
These parametric equations represent a line in R^3, which means that all three planes have a common line of intersection.
(b) An example of three planes in R^3 that intersect in pairs but have no common point of intersection is:
Plane 1: x - y + z = 0
Plane 2: 2x + y - 3z = 1
Plane 3: -4x - 2y + 6z = -2
To justify, we can again solve the system of equations formed by the planes. By solving the system, we find that the planes intersect in pairs as follows:
Plane 1 and Plane 2 intersect along the line:
x = t
y = -2t
z = t
Plane 2 and Plane 3 intersect along the line:
x = t
y = -2t
z = -t
Plane 1 and Plane 3 intersect along the line:
x = 0
y = 0
z = 0
Although the planes intersect in pairs, they do not have a common point of intersection because none of the lines intersect at the same point.
(c) An example of three planes in R^3 that intersect in a single point is:
Plane 1: x - y + z = 1
Plane 2: x + y - z = 2
Plane 3: 2x - y + 3z = 3
To justify, we can once again solve the system of equations formed by the planes. By solving the system, we find that the planes intersect at the single point (1, 0, 0).
Substituting these values back into the equations for each plane, we can confirm that all three planes intersect at this point:
Plane 1: 1 - 0 + 0 = 1
Plane 2: 1 + 0 - 0 = 2
Plane 3: 2 - 0 + 0 = 3
Since all three planes intersect at the point (1, 0, 0), they intersect in a single point.
(a) To find three planes in R^3 that have a common line of intersection, we can start by considering the xy, yz, and xz planes. The xy plane is the plane where z = 0, the yz plane is where x = 0, and the xz plane is where y = 0.
The equation of the xy plane is z = 0. The equation of the yz plane is x = 0. And the equation of the xz plane is y = 0.
If we solve these three equations simultaneously, we can find the common line of intersection.
Let's take the equations y = 0 and z = 0. If we substitute these values into the equation of the xz plane, we get y = 0 as well. This means that the point (x, 0, 0) lies on all three planes. Therefore, the common line of intersection is the x-axis, which can be represented as the equation x = t, where t is any real number.
So, the three planes in R^3 that have the common line of intersection x = t are the xy plane (z = 0), the yz plane (x = 0), and the xz plane (y = 0).
(b) To find three planes in R^3 that intersect in pairs but have no common point of intersection, we can consider the xy, yz, and zx planes.
The equation of the xy plane is z = 0, the equation of the yz plane is x = 0, and the equation of the zx plane is y = 0.
If we solve these three equations simultaneously, we can determine if there is a common point of intersection.
Let's take the equations x = 0 and y = 0. If we substitute these values into the equation of the xy plane, we get z = 0. This means that the point (0, 0, 0) lies on all three planes. Therefore, the three planes do have a common point of intersection, and they cannot be an example of three planes that intersect in pairs but have no common point of intersection.
So we need to find another set of three planes. Let's consider the equations x = 1, y = 1, and z = 1. If we substitute these values into the equations of the three planes, we get different constant values for each plane. This means that the three planes x = 1, y = 1, and z = 1 do not have a common point of intersection. However, any two of these planes do intersect in a line, for example, x = 1 and y = 1 intersect along the line (1, 1, t).
Therefore, the three planes x = 1, y = 1, and z = 1 in R^3 intersect in pairs but have no common point of intersection.
(c) To find three planes in R^3 that intersect at a single point, we can consider the following three planes:
Plane 1: x + y + z = 1
Plane 2: x + y + z = 2
Plane 3: x + y + z = 3
If we solve these equations simultaneously, we can determine if the planes have a common point of intersection.
Subtracting Plane 1 from Plane 2 and Plane 3, we get:
Plane 2 - Plane 1: x + y + z = 2 - 1 = 1
Plane 3 - Plane 1: x + y + z = 3 - 1 = 2
Now, subtracting Plane 2 from Plane 3, we get:
Plane 3 - Plane 2: x + y + z = 3 - 2 = 1
From these equations, we can see that all three planes intersect at the point (1, 0, 0) which satisfies each equation. Therefore, the three planes x + y + z = 1, x + y + z = 2, and x + y + z = 3 intersect at a single point (1, 0, 0).
It should be easy to come up with (c)
In fact, most of your exercises will intersect at a point.
Now just adjust one of the planes so that it works for (a) and (b)
I'm sure your text has examples of such situations.