Multi-Variable Calculus Assignment

due Friday, February 16, 2001

Exercises

1. Find the angle between the twisted cubics

s ↦ (s, s^{2}, s^{3}) and t ↦ (t^{2}, t^{3}, t)

at the point (1, 1, 1).

2. Find the angle between the line

t ⟼ (1 − 2t, −3 + t, 2 − t)

and the plane x + y + z = 0 .

3. Find the angle between the two planes x − 2y + z = 2 and 2x − 3y − z = 1 .

4. Let a > 0 and b be given constants, and let F be the parameterized helix

F (t) = (−a sin t, a cos t, bt) .

Let γ be the function defined by the statement that γ(t) is the arc length along the helix from the point F (0) to the point F (t). Find a formula for γ(t).

[Processed from GELLMU to L^{A}TEX on Tue 13 Feb 2001 at 10:14:45 EST]