110.202 CALCULUS III 7 October 1998

First Examination

11:00 a.m. Edition

50 minutes. Closed book. No notes. 80 points, 20 per question. Partial credit may be available, but only if you show your working. Begin each of the four questions on a new page and number it clearly in the margin. Make sure your T.A.’s name is on each book, as well as your name. Calculators are allowed but are not recommended or useful. Do not numerically evaluate square roots, trigonometric functions and such.

1. (a) Decompose the vector a = 4i + 2j − k as the sum c + d of two vectors, where c is parallel to the vector b = 2i + j + 3k and d is perpendicular (orthogonal) to b.

(b) Find the area of a triangle that has the vectors a and b as two of its sides.

2. A particle traces out the curve C. Its position at time t is given by

r(t) = t i + (3/2)t^{2} j + t^{2} k.

(a) Find its velocity at any time t.

(b) Find the coordinates of every point where C intersects the plane 2x+2y−z = 0.

(c) Find the angle between the curve C and the plane at one point of intersection (your choice) in (b).

3. The line l is given parametrically by the equation

r(t) = 2i − 2j + t(2i + 2j + k).

The point P_{1} has coordinates (7, 2, 3).

(a) Find an equation for the plane that contains the line l and the point P_{1}.

(b) Find the (shortest) distance between the point P_{1} and the line l.

4. A spring S is parametrized by the equation

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

Let P_{2} be the point of S where t = π.

(a) Find the unit tangent vector to S at P_{2}.

(b) Find a parametric equation for the tangent line to S at P_{2}.

YOU MAY RETAIN THIS QUESTION SHEET JMB