Lecture 4, Wednesday, 9/3/08:
Section 10.5: Equations of lines and planes in space
Topics
- Lines: vector equation,
parametric equations,
symmetric equations, direction vectors
- Planes: vector equation, linear equation, normal vectors
- Distance formulas:
-
Distance between a point and a line (covered in Section 10.4 of the book; see
formula on top of p. 822)
-
Distance between a point and a plane
(use the formula in the first line of p. 834)
- Intersections of lines/planes:
- Two lines: Given two lines, determine whether they
(a) are parallel (possibly the same), (b)
intersect at a single point, or (c) are skew (do not intersect and are not
parallel).
- Two planes: Given two planes, determine whether they (a) are
parallel (and possibly the same), or (b) intersect at a line.
- Line and plane: Given a line and a plane, determine
whether the line (a) lies entirely in the plane,
(b) is parallel to the plane (but does not lie in it),
or (c) intersects the plane at
a single point.
Read
Section 10.5.
Homework
Section 10.5: 3, 5, 7, 11, 15, 17, 20, 23, 27, 43, 45, 53, 63, 65.
Solutions.
Notes
- Lecture schedule:
Section 10.5 is a rather lengthy section; I didn't get through all of it
in today's lecture, so I plan to spend a good part of Friday's lecture
(9/5) to finish up by discussing planes. I will again put my own spin on
this material and set things up a little bit differently than is done in
the text. However, I recommend that you read ahead a bit to get the
Smith/Minton presentation of it, and be better prepared for Friday's
class.
Note that the second half of the above problems depends on the material
to be covered Friday.
- Note on vector equations:
Of the various types of equations for lines and planes, the vector
equations are the easiest to remember, and also the most intuitive given
the underlying geometry. Unlike most calculus books, the Smith/Minton
text does not explicitly mention vector equations (though the vector
equation for a line does come up a bit later, in Ex. 1.4 of Section
11.1). While you won't be directly asked to write down vector equations,
vector equations are a useful tool when dealing with lines and planes.
In fact, when asked to derive an equation for a line or a plane
(parametric, etc.), I recommend that you first derive a vector equation,
then write out this equation in coordinates to get the desired form of
equation.
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Last modified Mon 08 Sep 2008 09:41:33 AM CDT