Calculus II/Test 4
If we follow the previous course, Test 4 might include material on the power series.
- All the examples in Chapter 9.1 might be on the test (pp634-38)
- wikipedia:Dot product wikipedia:Cross product
- Vectors let's look at rotations.--Guy vandegrift (discuss • contribs) 16:56, 5 April 2017 (UTC)
- These examples from Chapter 9 look like good candidates for Test 4:
- Example 1 p664: Find the equation of a line (in x,y,z coordinates) in a given direction through a given point.
- Example 2 p665: Find the equation of a line (in x,y,z coordinates) through two points.
- Example 4 p667: Find the equation of a plane perpendicular to a given direction, passing through a given point.
- Polar coordinates Appendix H page A55
- Examples 2-5 are easy and at least one of these will be on the test.
- Example 6 is moderately difficult and will be on the test in one form or another.
- The whole class stopped at A58 and nothing on or after Example 7 will be covered.
Example 8 Challenge question "extra credit' and not a lot of it.[edit | edit source]
See p669 of textbook: Find the distance from a point to given a plane, if the plane is defined as follows:
- is some point on the plane (i.e., three given numbers).
- is normal to the plane (again, three given numbers).
Now draw the point and the plane in from a specific angle in which the given point and the normal lie in the plane of the paper (board), and use facts about dot-product. (I never bothered to define comp and proj as described on 652; I was familiar with comp but never heard of proj, but it is an easy concept to grasp if you understand this problem).
- Rule: If, is a unit normal to the plane, then the magnitude of is the distance from the point to the plane. In the figure:
- is any vector connecting the point to somewhere on the plane (the calculated distance should not depend on ).
- The unit normal is given by:
If we view the expression in curly brackets as some constant, and drop the subscript "1", then we have the formula for a plane normal to .
Note that the distance, if because the point is already on the plane.