Math 2210-1. Test 1. Fall 2007.

Name: September 17, 2007

Problem 1: /40

Problem 2: /20

Problem 3: /30

Problem 4: /30

Problem 5: /30

Total: /150

Instructions: The exam is closed book, closed notes and calculators are not allowed. Youare only allowed one letter-size sheet of paper with anything on it.You will have 50 minutes for this test. The point value of each problem is written next to the

problem - use your time wisely. Please show all work, unless instructed otherwise. Partial creditwill be given only for work shown.

1

2Problem 1(40=14+13+13). Let ~u = 2~i~j+~k and ~v = ~i+3~j+2~k be two vectors (startingat the origin).

(1) Find the cosine of the angle between ~u and ~v.(2) Find the coordinates of the four vertices of the parallelogram determined by ~u and ~v.(3) Find the area of the parallelogram determined by ~u and ~v.

3Problem 2(20=10+10). For the sphere

x2 2x+ y2 + 2y + z2 4z = 15,

(1) find the coordinates of the center P and the radius r;(2) find the equation of the plane tangent to the sphere at the point Q(0, 3, 0).

4Problem 3(30=15+15). Consider the position vector for the helix

~r(t) = (2 cos t)~i+ (2 sin t)~j + (5t)~k.

(1) Find the unit tangent vector ~T (t) and the acceleration ~a(t).(2) Find the normal and tangential scalar components of the acceleration: aN and aT .

5Problem 4(30=20+10).

(1) Find the parametric equations of the line of intersection of the two planes

x 2y + 4z = 14 and x+ y + 2z = 11.

(2) Find the point of intersection of this line with the plane x y + 2z = 1.

6Problem 5(30=20+10). Consider the curve z = 2y2 in the yz-plane.

(1) Sketch (carefully) and identify the graph of the surface obtained by revolving the curvearound the z-axis.

(2) Write the equation of the surface in cylindrical coordinates.