In this lesson, you will learn about planes by definition; determining parallel, orthogonal, intersecting planes, and finding equations (vector equation, general/scalar equation, linear/standard equation).
View all of the instructional videos, read the provided readings, and practice exercises/activities. These will help you master the objectives for this lesson. Pay attention to additional comments provided. Try to visualize the problems. In geometry, a theorem states that if a line is perpendicular to a plane, then it is perpendicular to every line containing in the plane. This explains why a normal vector of a plane is perpendicular to all vectors in the plane.
Upon completion of the lesson 4.1, you will be able to:
Determine equations of planes
Determine relative positions of planes
Determine the intersection of two planes
View all of the following instructional videos. These will help you master the objectives for this module.
Vector and scalar equations of a plane [3:21]
Source: dlippmanmath from YouTube
Determining the Equation of a Plane [9:59]
Source: bullcleo1 from YouTube
Equations of Planes: Parallel and Containing a Line [3:25]
Source: dlippmanmath from YouTube
Equations of Planes [13:51]
Source: MIT from YouTube
Intersections of Line and Planes Part 1 [9:33]
Source: AlRichards314 from YouTube
Example 2: Notice the normal vector of the given plane and the direction vector of the line are perpendicular (use the dot product to verify), this implies that the line is either in the plane or not on the plane but parallel to it. Use the point (-3,1,2) on the line, plug in the coordinates accordingly into the given plane equation and evaluate, you will get 0, which means that the point is in the plane. This is evident by 1) the dot product of the normal vector of the plane and the direction vector of the line is 0, and 2) the point (-3,1,2) lies both on the line and the plane. Thus, it implies that the line is in the plane and hence the intersection of the plane and the given line is the given line. Example 3: Apply the same concepts used in Example 2: the dot product of the normal vector of the plane and the direction vector of the line is 0, which could only mean two cases mentioned above. Use the point (-4,6,-1) on the line, plug in the coordinates accordingly into the given plane equation and evaluate, you will NOT get 0 this time, which means that the point is on the line but not on the plane. The two evidence together indicate the line is parallel to the plane and hence there is no intersection.The methods provided by the presenter are good, but it's also good to know there are alternative ways to solving the problems. Something to think about: how do you determine if a line is perpendicular to a given plane?
Determining the Angle Between Two Planes [7:23]
Source: bullcleo1 from YouTube
Determining the Distance Between a Plane and a Point [5:18]
Source: bullcleo1 from YouTube
This concept can be applied to the following two types of problems: 1) Find the distance between a line, parallel to the given plane, and the given plane. (Find a point on the line, and then find the distance between the point and the plane.) 2) Find the distance between two parallel planes. (Find a point on plane 1, then find the distance between the point and plane 2.)
Note: The distance is the same regardless of the point that you pick.
Intersection of Two Planes Part 1 [8:28]
Source: cheektowagafloe from YouTube
Intersection of Two Planes Part 2 [8:11]
Source: cheektowagafloe from YouTube
Intersection of Two Planes Part 3 [9:19]
Source: cheektowagafloe from YouTube
The presenter made a computational error about y at the end. The correct expression for y should be: y=-(1/3)t + (11/3). Note: There are different ways to find the line of intersection of two planes.
Intersections of Two Planes Part 1 [7:36]
Source: AlRichards314 from YouTube
Intersections of Two Planes Part 2 [5:22]
Source: AlRichards314 from YouTube
Line of Intersection of two planes by cross product [4:47]
Source: Wei Ching Quek from YouTube
Line as Intersection of 2 Planes [4:47]
Source: Linda Fahlberg-Stojanovska from YouTube
This video covers an example that is very similar to the previous video, except that a point on the line is not given. In order to find a point on the line of intersection, very often you would randomly set one of the variables, x, y, z to be equal to some value that you pick, so you may obtain a 2x2 system of equations in terms of the other two variable. If the 2x2 system has a solution, then you have a point on the line (like the example shown on the video). If the 2x2 system does not have a solution, then you will have to change the value that you picked and try again (it does happen). If you prefer the "cross product" method instead of the "elimination" method, I would recommend that you do the cross product of the normal vectors first to obtain a sense of the direction of the line, then decide what value to use for which variable may simplify the process of finding the coordinates of a point on the line. For example, if a line is not parallel to the xy-plane, then it will intersect the xy-plane at some point, and the z-coordinate of that point is 0.
Graphing a Plane on the XYZ Coordinate System [8:45]
Source: bullcleo1 from YouTube
Line as Intersection of 2 Planes [5:36]
Source: Linda Fahlberg-Stojanovska
Note: This video covers a example that is very similar to the previous video, except that a point on the line is not given. In order to find a point on the line of intersection, very often you would randomly set one of the variables, x, y, z to be equal to some value that you pick, so you may obtain a 2x2 system of equations in terms of the other two variable. If the 2x2 system has a solution, then you have a point on the line (like the example shown on the video). If the 2x2 system does not have a solution, then you will have to change the value that you picked and try again (it does happen). If you prefer the “cross product” method instead of the “elimination” method, I would recommend that you do the cross product of the normal vectors first to obtain a sense of the direction of the line, then decide what value to use for which variable may simplify the process of finding the coordinates of a point on the line. For example, if a line is not parallel to the xy-plane, then it will intersect the xy-plane at some point, and the z-coordinate of that point is 0.
The following required readings cover the content for this module. As you go through each reading, pay close attention to the content that will help you learn the objectives for this module.
The University of Sydney School of Mathematics and Statistics
Don't forget to view the next page.
Paul's Online Math Notes
Jim Lambers
Make your way through each of the practice exercises. This is where you will take what you have learned from the lesson content and lesson readings and apply it by solving practice problems.
Below are additional resources that help reinforce the content for this module.
Vector and Scalar Equations of a Plane (video) [3:21]
Source: dlippmanmath
Planes: Parallel, Equal, or Intersecting? (video) [3:55]
Source: MathDoctorBob
Finding the Scalar Equation of a Plane (video) [7:40]
Source: patrickJMT
Finding the Point Where a Line Intersects a Plane (video) [5:38]
Source: patrickJMT
Do Homework 4.1 on MyMathLab.
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