EXAMPLES FOR EXAM I

EXAMPLES FOR EXAM I

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--Vector Addition, Vector Properties

Vector Equilibrium Example

A 100-pound collar (the green rectangle) slides on a frictionless vertical rod. Find the distance y (the distance from A to B) for which the system is in equilibrium if the collar is attached by a pulley to a 120 pound counterweight.

y is the distance from A to B.

Vector Force Example--Tugboats

Click on the picture below to see an animation.

Distance Between Points in Space

Find the distance between points P and Q.

Click on the picture at the right to animate.

Finding The Equation Of A Sphere

DPGraphPicture Transparent Picture of the Sphere with Points P and Q

Finding The Center And Radius Of A Sphere

DPGraphPicture Transparent Picture of the Sphere with Center

(x - 1)² + y² + (z - 1)² = 27

Direction Cosines

Projections and Vector Components

The picture below represents the vectors in the example at the left with the colors corresponding to vectors as indicated in the picture above. Click on the picture below to see an animation.

Work Example

Click here to see the answer and a related animation.

Click here or on the picture above to see an animation. Click here for an animation without scales.

Work Example Similar To Section 10.3 #72 (6e 64)

Click on the picture at the right to animate the red force vector.

Example--Volume Of A Parallelepiped

DPGraphPicture

Click here to see an animation of the three vectors in standard position.

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 10.5 #38 (6e 34)

DPGraph Picture of the plane

Maple Picture of the plane and two vectors determining the plane

Example--Finding The Equation Of A Plane

Section 10.5 #40 (6e 36) 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.

Line Of Intersection Of Two Planes

DPGraphPicture1 DPGraphPicture2

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.

return

This site contains links to other Internet sites. These links are not endorsements of any products or services in such sites, and no information

in such site has been endorsed or approved by this site.

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