The dot product is involved in the computation for vector projection. "Vector projection"is involved, along with "cross product", in many distance finding problems in space. But first, we'll need to do some Analytic Geometry work in space. We'll develop equations for lines and planes. Instead of using the idea of slope, as in the case of work in a plane, we'll use a vector to determine direction for our lines and planes. It is important to remember that the equations we write for planes and lines are neither true nor false, rather they are "open" because they contain variables. The dichotomy of "true" versus "false" for the equations corresponds to the dichotomy of points on versus points off of the plane and/or line.
Notice that the authors have included the proofs for Thm.11.11 and for the equivalence of the symmetric equations and the parametric equations without labeling them "Proof". In the proof of Thm 11.11 the subcripted variables x,y and z represent the coordinates of a particular point in space while the triplet of variables x,y,and z represent the coordinates of any point in space; that, of course, would include points on the line as well as off the line. Then the equations are neither true nor false but are "satisfied" only by coordinates of points on the line; the coordinates of points off the line "cause" the equations to be false---there you have the dichotomy.
In the explanation for the symmetric equations you need to realize that elimination of the parameter,t, can easily be accomplished by solving each of the parametric equations for t and them equating the results.
Examples of the use of the Thm.11.11 follow:
See 11.5 top of Pg.799(equations)
Next we consider planes in space. The first crucial idea is that a line in the plane will not "determine" the direction of the plane; in our previous work on line, a vector on the line did "determine" the direction of the line. Instead, we use a vector perpendicular to the plane to "determine" the direction of the plane.
If you place a pencil perpendicular to your hand, you can "see" the effect of change of direction of the pencil on the direction of you hand; your hand represents the plane and the pencil represents a vector.
Your authors' work on planes, given that we've done the Analytic Geometry for the line, should now be fairly clear. Again, we recognize that the proofs are not explicity referred to but rather included in his writings about the Thm.11.12 and equation.
An example of the use of Thm.11.12 follows:
The next example from the text solves two problems that would "logically" occur when two planes intersect, namely, "What is the "size" of the angle of intersection?" and "Find the line of intersection of the two planes."
See 10.5 following Example 4 ( the vector result )
The graph of a plane can easily be drawn as follows:
See 11.5 Sketching Planes in Space
This "completes" our work on the Analytic Geometry for lines and planes. It was a fairly lengthy lesson and you may wish to take a "break" before we look at the application problems that can be done using dot and cross products along with the equation work we just finished for lines and planes.
Distance Between Points, Planes, and Lines
See 11.5 Theorem 11.13 Pg.803 I want you to have this theorem first because it results in a second formula that your author uses to find distance from a point to a plane. If we "say" this theorem it would be something like: "Compute the absolute value of the dot product of a vector between a point in the plane and the given point off the plane with the normal vector for the plane and divide by the norm of that normal vector." This computation will result in the measure of the distance from the given point and the plane( of course, here we mean the shortest distance ).
See 11.5 distance between points, planes, and lines
The distance formula that follows from Thm.11.13 would "say" something like: "The absolute value of the evaluation at the coordinates of the given point for the "left side" of the equation in the plane divided by the norm of the normal vector for the plane results in the measure of the distance from the point to the plane.
When your authors and I say that this results from Thm.11.13 we do not mean that it is obvious and it may require some collaboration.
For finding the distance from a point to a line, it is "neat" that we just replace the dot product by the cross product and the normal vector with a vector that lies along the line in the formula in Thm.11.13. The Proof of this does not "employ" dot product or cross product but rather uses "right triangle trigonometry".
See 11.5 Thm.11.14 (and proof)
Examples of distance finding problems follow:
This next example may help you with an idea that your author addressed above, but one that may have confused you. It has to do with finding the coordinates of a point in a plane given that you have the equation for the plane. The authors explained it with equation work but perhaps it will help if we "say" the idea: "To get the coordinates of a point in a plane whose equation is given, we assign zeros to two of the three variables in the equation and then solve for the numerical value for the third variable." This idea is now illustrated in this next example.
Lastly, an example of finding distance from a point to a line.
Look at the following solved problems
in the Exercises at the end of
11.5 : 1,5,7,9,11,13,15,25,27,29,33,3541,57,59,65,67
Then do the following unsolved
problems as homework: 8,10,12,16,26,28,30,58,60,64,68