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CalcVR Supplemental Materials

Jeremy Becnel, Nicholas Long

Section 6.2 Change and Slopes on Non-vertical Planes

In this lesson, the user explores nonvertical planes (planes that can be expressed as \(z=f(x,y)\)) then looks at how change in a vertical coordinate can be measured in terms of the horizontal coordinates. In the second lesson, this idea is expanded to produce a measure of change in directions that are not parallel to the \(x\) or \(y\) axes and to develop the point and slopes form of non-vertical planes.

Subsection 6.2.1 Vertical Change on a Plane

Activity 6.2.1.

A non-vertical line in space is said to be in the \(y\) direction if it is on a plane of the form \(x=c\) for some constant \(c\text{.}\) A non-vertical line in space is said to be in the \(x\) direction if it is on a plane of the form \(y=c\) for some constant \(c\text{.}\)
(a)
On the plane that is below on the left, darken three lines that are in the \(x\) direction. You can get a larger version of each plot by clicking on each image. You can also download each image separately and print your own copy.
On the plane that is below on the right, darken three lines that are in the \(y\) direction.
Figure 6.2.1.
(b)
In EACH of the following planes identify a line in the \(x\) direction and find its slope. You should choose a line with two points whose coordinates may be easily determined. You should state or label each of the points you are using to find the slope of a line in the \(x\) direction.
Figure 6.2.2.
(c)
In Each of the following planes identify a line in the \(y\) direction and find its slope. You should choose a line with two points whose coordinates may be easily determined. You should state or label each of the points you are using to find the slope of a line in the \(y\) direction.
Figure 6.2.3.
(d)
Let P be the plane with equation \(z=2x+3y+4\text{.}\)
  1. The intersection of P with the fundamental plane \(x=5\) is a line with slope \(m=\text{?}\) Breifly explain how you got your slope.
  2. Reflect on what you did in the previous part and explain why in general all the lines on a plane, that are in the \(y\) direction (this means that \(x\) is constant on the line) have the same slope.
  3. The intersection of P with the fundamental plane \(y=3\) is a line with slope \(m=\text{?}\)
  4. Reflect on what you did in the previous part and explain why in general all the lines on a plane, that are in the \(x\) direction (this means that \(y\) is constant on the line) have the same slope.
(e)
A plane contains points given by the following table. The entries in the table are the \(z\) values for the given \(x\) and \(y\) values. For instance, if \(x=1\) and \(y=4\text{,}\) then the \(z\)-value for the plane is 3 (look in the first row of x values and the second column of y values.)
Table 6.2.4.
y=2 y=4 y=6
x=1 5 3 1
x=2 8 6 4
x=3 11 9 7
  1. Find the slope in the \(x\) direction of the plane.
  2. Briefly explain why the same slope will be obtained regardless of what two points are used to compute it.
  3. Find the slope in the \(y\) direction of the plane.
(f)
This problem refers to the plane in the following figure. Recall that on a line: \(\text{vertical change} = \text{(slope)} \times \text{(horizontal change)}\text{.}\)
  1. If you start at point \((0,0,1)\) of the above plane and move on the plane in the \(x\) direction in such a way that \(dx\) (the horizontal change in \(x\)) is \(2\text{,}\) find \(dz_x\) (the vertical change in the \(x\) direction).
  2. On the above graph, darken and label a horizontal segment that represents dx (since the segment is horizontal it will NOT be on the plane) and a vertical segment that represents \(dz_x\text{.}\)
  3. If you now continue moving on the plane, but this time in the \(y\) direction in such a way that \(dy\) (the horizontal change in \(y\)) is 2, find \(dz_y\) (the vertical change in the \(y\) direction).
  4. On the above graph, darken and label a horizontal segment that represents \(dy\) (since the segment is horizontal, it is NOT on the plane) and a vertical segment that represents \(dz_y\text{.}\)
  5. Use the fact that on a line, the vertical change is the slope multiplied by the horizontal change to complete the following table, one line at a time.
    Table 6.2.5.

    Initial
    Point    		 Horizontal change
    in the \(x\) direction,
    \(dx\)     		Vertical change
    in the \(x\) direction,
    \(dz_x\)     		Horizontal change
    in the \(y\) direction,
    \(dy\)     		Vertical change
    in the \(y\) direction,
    \(dz_y\)     		Total vertical
    change
    (0,0,1) 3 5
    (3,4,12) -4 2
    (a,b,c) 2 -1

Activity 6.2.2.

(a)
Use the notion of vertical change on a plane \(dz\) to find the equation of the plane represented in the following table
Table 6.2.6.
y=0 y=3 y=6
x=2 18 20 22
x=4 15 17 19
x=6 12 14 16
(b)
Use the notion of vertical change on a plane \(dz\) to find the equation of the plane represented in the following figure:

Activity 6.2.3.

(a)
In this problem, the slope of the line segment \(\overline{PQ}\) is computed, first in the direction of the vector \(\overrightarrow{v}\) given in the figure and then in the direction of \(-\overrightarrow{v}\text{.}\) Fill in the table below.
Table 6.2.7.
Direction \(\Delta V\) \(\Delta H\) Slope
\(\vec{v}\)
\(-\vec{v}\)
(b)
Fill the table below based on the graph below.
Table 6.2.8.
Direction \(\Delta V\) \(\Delta H\) Slope
\(\vec{v}\)
\(-\vec{v}\)
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