In this activity we use a simple illustration of a plane to recognize a property of planes that can be used to define them.
(a)
Use a textbook or similar object to represent a plane. Take a pencil or similar vector like object and place it so that it is orthogonal (perpendicular) to the textbook (plane).
(b)
Place a second vector object on the plane at a different point on the plane but also orthogonal to the plane. Compare the direction of the two orthogonal vectors.
(c)
Repeat this experiment with a non-planar surface. You can use a backpack, your leg, or many other items. Contrast the results to those of the plane.
Activity13.
Here we use the property from Activity 12 to generate an equation for planes.
Use \(\vec{v}=(11,13,7)\text{,}\)\(P_1=(8,4,5)\text{,}\)\(P_2=(7,7,1)\text{,}\)\(P_3=(11,2,4)\text{,}\) and \(P_4=(9,8,-4)\text{.}\)
(a)
First we confirm that these points are all in the plane.
(i)
Construct \(\vec{u}_1=\stackrel{\longrightarrow}{P_1P_2}\text{.}\) Confirm that \(\vec{v} \cdot \vec{u}_1 = 0\text{.}\)
(ii)
Construct \(\vec{u}_2=\stackrel{\longrightarrow}{P_1P_3}\text{.}\) Confirm that \(\vec{v} \cdot \vec{u}_2 = 0\text{.}\)
(iii)
Construct \(\vec{u}_3=\stackrel{\longrightarrow}{P_1P_4}\text{.}\) Confirm that \(\vec{v} \cdot \vec{u}_3 = 0\text{.}\)
(b)
Let \(P\) and \(Q\) be any two points in the plane. What do we know about \(\vec{v}\) and \(\stackrel{\longrightarrow}{PQ}\text{?}\)
(c)
Using \(\vec{v}\text{,}\)\(P_1\text{,}\) and an arbitrary point in the plane \(Q=(x,y,z)\) write an equation for this plane.
Activity14.
In the previous activity we generated one form of an equation for a plane. In this activity we consider what information we need to generate that equation for any plane.
(a)
In the previous activity we used one vector orthogonal to the plane and how many points in the plane? Reference only the last step.
(b)
If we have only three points in the plane (no vector orthogonal to the plane), how can we generate an equation for this plane?
Subsection1.3.2Lines and Planes
Watch the video in Figure 1.3.1 to learn how to sketch 3D surfaces before continuing. Note sketching in 3D by hand will lead to a better understanding of functions and surfaces. We learn it to understand them rather than because graphing by hand is a useful skill for daily life. This section illustrates relationships between curves—including lines—and surfaces—including planes.
Figure1.3.1.Demonstration of Graphing 3D Surfaces by Hand
Table1.3.2.Example Table of Values
x
-2
-1
0
1
2
-2
-1
y
0
1
2
Activity15.
By graphing a plane we will begin to understand how to analyze surfaces.
(a)
Complete a table like Table 1.3.2 for the plane \(2x+4y-6z+2=0\text{.}\)
(b)
Graph the points with \(y=-2\text{.}\) What is the shape you graphed?
(c)
Graph the points with \(y=-1\text{.}\) What is the shape you graphed?
(d)
Graph the points with \(x=2\text{.}\) What is the shape you graphed?
(e)
In general a plane consists of what (based on previous three answers)?
Activity16.
In this activity we learn to express planes with a parametric form.
For this activity use \(\vec{a}=(3,1,1),\)\(\vec{b}=(2,-5,-3),\) and \(C=(1,0,1).\)
(a)
Confirm this is a plane by graphing using technology. If you do not have a preferred tool, Wolfram Alpha will produce a graph if you enter this equation.
(b)
Next we confirm this is a plane by converting it to the previous notation.
(i)
Use this equation to generate three points in the plane.
(ii)
Use these points to generate two vectors.
(iii)
Use the vectors to generate an equation for the plane.
(iv)
Calculate \(\vec{a} \times \vec{b}\text{.}\) Use this to write an equation for the plane. Is it the same plane as the previous equation you generated?
(c)
Recall that the equation of a line in any dimension is
\begin{equation*}
X = \vec{m}t+B\text{.}
\end{equation*}
Be prepared to explain why
\begin{equation*}
X = \vec{a}t+\vec{b}s+C
\end{equation*}
makes sense as the equation of a plane.
(d)
Compare the planes
\begin{align*}
X = & (1,1,0)t+(0,1,1)s+(8,2,6)\\
X = & (1,1,0)t+(0,1,1)s+(1,2,4)
\end{align*}
What is the difference between these? How can we interpret \(C\) in the formula for plane?
Orthogonality provides solutions to many problems. The distance between a point and a line or plane, the distance between non-intersecting lines, the distance between a line and a plane, and similar distances are defined as the shortest distance between some point on one and some point on the other. It can be shown (and will be later) that the distance between a point and a line or a plane is the length of a segment from the point that is orthogonal to the line or plane.
Activity17.
Use the following to demonstrate that this also holds true for the distance between parallel lines or parallel planes.
(a)
Find the distance between the parallel lines \(\ell_1(t)=(1,3,1)t+(1,2,2)\) and \(\ell_2(t)=(1,3,1)t+(2,2,3).\)
(b)
Find a plane such that every point on the plane is equidistant from the points \((1,3,1)\) and \((1,2,2).\) Note the 2D version of this was a theorem in high school geometry.
