Change and Slopes on Non-vertical Planes

Section 8.2 Change and Slopes on Non-vertical Planes

Objectives

To look at how to measure change in the vertical coordinate in terms of change in the horizontal coordinate for nonvertical planes
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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.

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Subsection 8.2.1 Vertical Change on a Plane

Activity 8.4.

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 \(c\text{.}\)

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(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.

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On the plane that is below on the right, darken three lines that are in the \(y\) direction.

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(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.

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(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.

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(d)

Let P be the plane with equation \(z=2x+3y+4\text{.}\)

The intersection of P with the fundamental plane \(x=5\) is a line with slope \(m=\text{?}\) Breifly explain how you got your slope.

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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.

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The intersection of P with the fundamental plane \(y=3\) is a line with slope \(m=\text{?}\)

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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.

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(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 8.5.

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	y=2	y=4	y=6
x=1	5	3	1
x=2	8	6	4
x=3	11	9	7

Find the slope in the \(x\) direction of the plane.

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Briefly explain why the same slope will be obtained regardless of what two points are used to compute it.

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Find the slope in the \(y\) direction of the plane.

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(f)

This problem refers to the plane in the following figure. Recall that on a line: \(\text{vertical change} = \text{(slope)} \times \text{(horizontal change)}\text{.}\)

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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).
  
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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 \(dz_x\text{.}\)
  
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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).
  
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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 \(dz_y\text{.}\)
  
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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. The spaces labeled with * are filled with numbers and the ones labeled with ** are filled with an expression in the variables \(dx\text{,}\) \(dy\text{.}\)

Table 8.6.

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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)	dx	**	dy	**	**

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Explain in your own words why the formula \(dz=dz_x+dz_y\) makes sense.

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Explain in your own words why on a plane it is true that \(dz = m_x dx + m_y dy\text{.}\)

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Reflect on what you did on the previous parts and then explain in your own words how the vertical change from one point on the plane to another point on the plane may be found if you don’t know the \(z\) coordinate of the final point but you know the change in \(x\text{,}\) the change in \(y\text{,}\) and the slopes in the \(x\) and \(y\) directions.

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Explain in your own words why the expression for total vertical change in the third row of the above table does not depend on the initial point \((a, b, c)\text{.}\)

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Activity 8.5.

(a)

Use the notion of vertical change on a plane \(dz\) to find the equation of the plane represented in the following table

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Table 8.7.

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	y=0	y=3	y=6
x=2	18	20	22
x=4	15	17	19
x=6	12	14	16

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(b)

Use the notion of vertical change on a plane \(dz\) to find the equation of the plane represented in the following figure:

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Activity 8.6.

(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.

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Table 8.8.

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Direction	\(\Delta V\)	\(\Delta H\)	Slope
\(\vec{v}\)
\(-\vec{v}\)

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(b)

Fill the table below based on the graph below.

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Table 8.9.

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Direction	\(\Delta V\)	\(\Delta H\)	Slope
\(\vec{v}\)
\(-\vec{v}\)

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