EXAMPLES FOR EXAM I

EXAMPLES FOR EXAM I Section 11.5

If it is not already on your hard drive, you will need to download the free DPGraph Viewer to view some of the pictures linked to on this page.

QuickTime 7 free download.

Here is a Maple Worksheet with dot product and cross product examples.

Example--Finding The Equation Of A Line

If we change the point to P(7,6,3) then the set of parametric equations for the line will be

x = 7 + 3t

y = 6 + 2t

z = 3 - t

with the corresponding graph the lower picture on the right.

Example--Intersecting Lines

Click the picture above to see an animation.

Maple Worksheet Demonstrating Various Planes

Example--Finding The Equation Of A Plane

Click DPGraphPicture to see a DPGraph picture of the plane.

Example--Finding The Equation Of A Plane

Section 11.5 #46

DPGraph Picture of the plane

Maple Picture of the plane and two vectors determining the plane

Example--Finding The Equation Of A Plane

Section 11.5 #48 DPGraph Picture of the Plane

Click on the picture above to see an animation.

Extra Credit: This link will take you to an Intermediate Algebra powerpoint presentation on Solving Systems of linear equations. Beginning with slide 40 there is a brief College Algebra level introduction to solving systems of three linear equations (whose graphs would be planes). You can receive extra credit for doing the extra credit presented at the end of the powerpoint presentation. Same powerpoint presentation beginning with slide 40

Example--Distance Between A Point And A Plane

Find the distance from the given point Q to the given plane (1). DPGraph Picture You must animate the DPGraph Picture to see the point.

Example--Distance Between Two Parallel Planes

DPGraphPicture

A More General Example--Two Adjustable Parallel Planes

DPGraphPicture--Adjustable

Two Planes That Can Be Made To Intersect

DPGraphPicture--Intersection DPGraphPicture2

Example--Distance Between Two Skew Lines

Click on the picture to see an animation. The distance is equal to the distance between the following two parallel planes.

x - y + 2z = 7

x - y + 2z = 1

Click DPGraphPicture to see the two parallel planes.

Example--Distance From A Point To A Line In Space

Two Solution Methods Given Below

Click on the picture to see an animation.

Example--Distance From A Point To A Line In Space

(with a WinPlot demonstration)

Find the distance from the point P(3,4,5) to the line given parametrically by

x = 12 - 2t

y = 3t

z = 2t

Click here to see a demonstration using Winplot. You may need to download the file to your desktop and then use the freeware Winplot to open the file (by opening Winplot, clicking on Window, clicking on 3-dim, clicking on File, clicking on Open, and then opening PointToLine. You can use the slider to vary the value of A from 0 to 6. This will move a point along the line and give the distance from P(3,4,5) to the point. You can observe that the point on the line closest to P is about 5.28 units from P. You could also activate sliders for B, C, and D from the "Anim" menu to move point P(B,C,D).

Line Of Intersection Of Two Planes

DPGraphPicture1 DPGraphPicture2

Bonus: Find an equation of a plane containing the line of intersection of the two planes above and construct a DPGraph picture of the three planes.

The Acute Angle Theta Between The Two Planes

(The Acute Angle Between Their Normal Vectors)

Intersection of a Line and a Plane

Extra Credit

This DPGraphPicture shows the graphs of two planes whose equations are

There is something in the picture that relates to the problem above beyond the fact that one of the planes is involved

in the problem above. Figure out what that is and prove it.

Distance Between Skew Lines

When I demonstrate the TI-89/92 program Dist3D (which will compute the distance between two points in space, a point and a line in space, a point and a plane, two parallel planes, two parallel lines, and two skew lines) we observe a perhaps unexpected result in calculating the distances between some pairs of skew lines. Here is the analytical justification for what happened.

Thus the distance is independent of c and d as seemed to be the case based on the TI-92 program results.

Another interesting result is that the two parallel planes containing the skew lines, the distance between which is the distance between the skew lines, do not change when c and/or d change. Their equations are shown below.

Extra Credit: Prove a similar result for

That is, prove that the distance between these two lines is independent of a and d.

Example: a = 1, b = 4, c = 5, d = -1

DPGraph Picture of the Planes

Below is a picture of the two lines with d = -1 and c = 5 as above. Click on the picture to see an animation. Click here to see an animation relating to the two lines but with d varying from -4 to 2 as c varies from 2 to 8. The distance between each pair of lines in the animation is the same and is the distance computed above. The two lines remain in the same two planes pictured in the DPGraph Picture, one in each.

Click here to see an animation of the changing lines within the two parallel planes.

Click here to see the same animation with the line segment representing the distance between the two lines included. Line Segment Endpoints' Paths In Red Only One Plane Showing Zoom On Red Paths

Click here to see the animation with the scaling constrained so that the orthogonal relationship is more apparent.

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Lane Vosbury, Mathematics, Seminole State College email: vosburyl@seminolestate.edu