Problems are from the text (EP=Edwards and Penney) or from the supplementary notes (SN).
Hand in underlined problems from Part I and all of Part II. Note that the solutions to Part I problems are generally available in the notes, Section S. Part II is marked more critically with points as indicated. These points will accumulate and finally constitute 30% of the possible total.
Lec 5 (Thurs Feb 11): Read EP Sect 12.4, 10.4 to p. 591, 12.5 to p. 747. Problems:EP p. 742 nos. 3, 7, 22, 33. EP p. 594, nos. 4, 12, 15. EP p. 755 nos. 3, 4, 13, 31, 39, 40.
Lec 6 (Fri Feb 12): Read EP Sect 12.6, SN K. Problems SN p. K.2, nos. 1, 2, 3.
Problem 1: (3 pts) Given that and find all vectors of length which are perpendicular to both and
Solution: The cross product is This has length so the two vectors of length perpendicular to both and are
Problem 2: (2+2 pts) Let be the origin, and let be the line through the origin parallel to
a) Express the vector as the sum of a vector parallel to and a vector perpendicular to
b) Calculate the distance from to
Solution:
a) and is parallel to the line. Since has length the vector has the same dot product with as has (namely Thus
b) A general point on the line is (since it is plus some multiple of The length of this is greater than or equal to which is therefore the distance from the origin to the line.
Problem 3: (2+2 pts)
Consider the system
a) For what values of the constant will there be a non-trivial solution?
b) Let Find a non-trivial solution by writing the three equations in vector form as and then using vector analysis to find a non-zero vector which is orthogonal to all three vectors and
Solution:
a) The determinant of the matrix is
b) If then the equations are and if and The cross product of and is - which is orthogonal to all three. Thus a non-trivial solution is and
Problem 4: (8 pts)
Does a pitched ``curve ball'' in baseball really curve? Work through
project 12.5 in EP. There a several questions to be answered; in your
answer label then (a), (b) and so on, in order. Write up the last two in
decent English.