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PRACTICE EXAM I
Math 237, Spring 2008
Answer each of the following problems as completely as you can. You must show your work to receive credit. Progress in the right direction will be worth partial credit. THIS IS A PRACTICE EXAM. In particular, it is not an exhaustive list of topics that are worth reviewing. The algebra of these problems, and typos generally, are not vetted as carefully as they would be on a real test. It is also a little longer than the real test will be.
(1) (a) What is the definition of (a, b)?
(b) How might one compute (a, b) for vectors in R^3?
(c) Identify and sketch the surface: x^2 = 3y^2 + 4z^2 Label co-ordinate axes and intercepts.
1
(2) Suppose that a = 〈 2 , − 1 , 0 〉 and that b = 〈 4 , 3 , 7 〉.Compute each of the following: (a) compab
(b) The area of the parallelogram with sides a and b.
(c) The equation of the plane that contains both vectors and the point (13, 13 , 42).
(4) (a) Find the distance traveled along a circular helix centered around the z-axis from the point (1, 0 , 0) to the point (1, 0 , 4 π), if that corresponds to two complete revolutions.
(b) Give an arc length parametrization of this curve.
(5) Let P = (1, 0 , 2), Q = (− 2 , 1 , 2), and R = (0, − 1 , −1). Are P, Q, and R co-linear? If so, what is an equation of the line that contains them? If not, what is an equation for the plane that they determine?
