SUNY Geneseo Department of Mathematics
Friday, February 24
Math 223
Spring 2023
Prof. Doug Baldwin
Homework 4, question 3 (given a plane’s normal, and points P and Q in the plane, find another point in the plane but not colinear with P and Q)?
At least 2 ways you can do this: For a more brute force way, find the equation for the plane (using the normal vector and one of the points), and the equation for the line through the points (using the difference of the points as the direction vector, and either point as the reference point). Then make up points that satisfy the plane equation until you get one that does not satisfy the line’s equation.
A more elegant way is to find the cross product of the plane’s normal and the vector between the points. This vector is perpendicular to the normal and thus parallel to the plane, so adding it to any point in the plane will give another point in the plane. The cross product is also perpendicular to the line through P and Q, so adding it to either of those points will give a point not on the line. Together, these two observations mean that adding the cross product to point P (or Q) is guaranteed to result in a point that’s in the plane but not colinear with P and Q:
For homework 4, question 4 (where do a line and a plane intersect) it would be best to give the actual point of intersection, not just the parameter value for the line’s equation.
For homework 4, question 5 (how to tell if a line and plane are parallel), the key thing is to find a normal for the plane. Then the line and plane are parallel if the line and normal are perpendicular, which you can determine by checking if their dot product is 0:
Next Tuesday (February 28) is Diversity Summit day.
There are no classes, in the hope/expectation that everyone will use the time to go to Diversity Summit events. Please do.
This means we have no class Tuesday.
Also in that spirit I’m discouraging but not totally refusing appointments that day.
There’s another math colloquium coming up.
Next Wednesday, March 1, 4:00 - 5:00 PM, Newton 204
Carolyn Cronauer (Geneseo class of ’09), Rochester School of the Deaf
“Math Talk for Deaf Students”
On teaching math to deaf students, particularly how American Sign Language evolves to express mathematical ideas.
Based on “Integrals of Vector-Valued Functions” in section 2.2.
An integral (either definite or indefinite) of a vector-valued function is just a vector of integrals of the components of the function.
Constants of integration become constant vectors.
Examples of definite & indefinite integrals?
As an indefinite integral, consider the antiderivative of r(t) =⟨3t2 - 3/t2, 5/(5t+2), sin t cos t ⟩. Start by integrating each component:
Now make a vector from each of those component integrals:
To see how the constant of integration becomes a vector, the “+ C” terms in each component integral can be collected into a vector that’s added, by vector addition, to the vector of specific antiderivatives:
For an example of a definite integral, consider integrating s(u) = ⟨u2 - 2, u cos(u2) ⟩from -1 to 1. Here also, start by finding antiderivatives of each component.
Then evaluate the antiderivatives at the upper and lower bounds, and make a vector from the resulting numbers:
Problem set 5, on vector-valued functions and their calculus, is ready.
Work on it next week, grade it the week after.
See the handout for more information.
Answer a remaining question about integrals of vector-valued functions involving dot or cross products.
Look at how to do calculus with vector-valued functions in Mathematica.
Time permitting, look at the question of how long a curve in space is.
Please look at “Arc Length for Vector Functions” and “Arc-Length Parameterization” in section 2.3 of the textbook, although we most likely won’t fully talk about that material until next Wednesday.