- Here's a link to the course syllabus.
- Our first paper homework set had two problems. In the first problem, the task was to find the point nearest (a,b,c) on the line through (1,0,0) and (2,1,1), where (a,b,c) are the last three digits of your own UIN.
The answer is found by observing that the vector v parallel to the line is (1,1,1) and the vector u from (1,0,0) to (a,b,c) is (a-1,b,c). So the projection p of u onto v is ((a-1,b,c)dot(1,1,1))/3 times (1,1,1), which works out to (a+b+c-1)/3 times (1,1,1). Add that to (1,0,0) to get the answer. For my UIN, p=(14/3,14/3,14/3) and the nearest point is (17/3,14/3,14/3). Different UINs will result in different details, of course.
The second problem was to find the point in a certain plane and nearest to (2,3,4). The plane is the plane through (0,0,0), (1,1,1), and (1,3,2).
Two vectors parallel to the plane are (1,1,1) and (1,3,2), and if we needed a third we could take the vector from (1,1,1) to (1,3,2) and get (0,2,1).
The cross product of any two of these will give a vector n perpendicular to the plane. So taking the first two of our vectors, we'd get n=(-1,-1,2).
The vector from (0,0,0) to (2,3,4) is just (2,3,4) and the projection p of (2,3,4) onto n is (3/6)(-1,-1,2)=(-1/2,-1/2,1). The point we seek, call it q, is also the vector v from (0,0,0) to q. Now v has two properties: v+p=(2,3,4), and v.n=0. The first property gives us v: v=(2,3,4)-p=(5/2,7/2,3). The second property is a check: is (5/2,7/2,3) dot (-1,-1,2) really 0? And yes, it is. So we are happy and (5/2,7/2,3) is both calculated and checked to be the answer to the question.
- Here's a sheet about quadric surfaces .
- New homework, due Monday Sept 14:
Problem 1: Sketch the quadric surfaces z=x^2+y^2 and z=2x^2+3 y^2-1. Is the intersection an ellipse? Is the projection of the intersection onto the x-y plane an ellipse?
Problem 2: Sketch the curves r1[s]=(s cos[2 pi s], s sin[2 pi s],s^2) and r2[s]=(s cos[6 pi s],s sin[6 pi s],s^2) over the interval s from 0 to 2.
There is a quadric surface that both curves lie on. Which sort of surface is it? (Give the name). What is an equation for the surface?
- Here's a sheet graphing 3D curves. .
about graphing curves in space.
- Here's solutions to the second homework .
- Here's the 3D pictures, contour pictures, and underlying functions featured during the Sept 16 lecture.
- Here's the PDF version of the paper homework assigned for Sept 21.
- Here's the sheet (also emailed to the class) with examples of limit problems.
- Here's solutions to both editions of the exam of September 30. Section 519 and Section 520.
- Here's solutions to both editions of the exam of Dec 3. Section 519
and Section 520.