Name: Calculus III
Exam 1
4 October 2000



My mind rebels at stagnation. Give me problems, give me work ... and I am in my own proper atmosphere.
-Sherlock Holmes from The Sign of the Four by A. C. Doyle.


Instructions: Show all work to receive full or partial credit. You may use a scientific calculator/graphing calculator. All University rules and guidelines for student conduct are applicable.

1.

[15 pts] If

$$\begin{eqnarray*}\v a & = & \langle5, -1 \rangle, \\ \v b & = & \langle10, 1 \rangle, \\ \v c & = & \langle-1, 2 \rangle, \\ \end{eqnarray*}$$

find scalars k and l such that

$$k \v a + l \v b = \v c.$$

What is the angle between $\v a$ and $\v b$?

2.

[20 pts] If

$$\v r(t) = e^{-t} \cos(t) \widehat{i} + e^{-t} \sin(t) \widehat{j} + t \widehat{k},$$

find the unit tangent vector, $\v T$, and the curvature, $\kappa$ when t=0. What is the radius of curvature at this point?

3.

[15 pts] Which two of the four parametrized lines below represent the same line?

$$\begin{eqnarray*}\v r_1(t) & = & \langle2, 1, 0 \rangle+ \langle1, -\frac{1}{2},... ...r_4(t) & = & \langle2, 0, 1 \rangle+ \langle-4, 2, 16 \rangle t \end{eqnarray*}$$

4.

[20 pts] Find the distance from the point (-2, 1, 3) to the plane

x+y+2z=1.

5.

[15 pts] Find the area of the triangle with vertices (-3, -2, -2), (2, 3, -2) and (1,-2,2).

6.

[15 pts] Find the equation of the plane containing the points (1,1,1), (2,1,3) and (-1,4,5).