Math 113 - Worksheet 1 - 9/4/2015
Instructions:
• Pair up with one or two (preferably two) other people in the class (at least one of whom
you don’t know).
• Pick one of the problems below to work on. (You should all work on the same problem).
• Read the problem and discuss possible approaches.
• Agree on an approach as a group. [If you have trouble, ask me.]
• Carry out the approach individually - if you get stuck, ask another member of your
group. [If you don’t get sufficient enlightenment from within you group, ask me.]
• When all the members in your group have finished, check your answers to see if they
agree, and if they are reasonable. Also, check with me to see if your answer is correct.
[ The goal of this worksheet is to give you extra practice with the material from Chapter 12.
You don’t need to complete the problems or turn them in. ]
1. Decide if each of the statements are true or false. Make sure that everyone in your group
agrees.
(a) Suppose ~v1 , ~v2 and ~v3 are three vectors in R2 . If ~v1 is perpendicular to ~v2 , and ~v2 is
perpendicular to ~v3 , then ~v1 is parallel to ~v3 .
(b) Suppose ~v1 , ~v2 and ~v3 are three vectors in R3 . If ~v1 is perpendicular to ~v2 , and ~v2 is
perpendicular to ~v3 , then ~v1 is parallel to ~v3 .
(c) Suppose ~v1 , ~v2 , ~v3 and ~v4 are four vectors in R3 . If ~v2 and ~v3 are perpendicular, and ~v1 and
~v4 are each perpendicular to both ~v2 and ~v3 , then ~v1 and ~v4 are parallel.
2. Let ~v = h3, 4i and let w
~ = h1, 2i. Find vectors ~a and ~b so that ~v = ~a + ~b so that ~a is parallel
to w
~ and ~b is perpendicular to w.
~
3. Find an equation for the plane containing the points (1, 1, 1), (1, 2, 3) and (1, 3, 6).
4. Which point on the plane x + 2y + 2z = 6 is closest to the origin?
1
2
5. Consider the surface z = Ax2 + Bxy + Cy 2 (and assume A 6= 0).
(a) Factor the equation as z = A(x2 + B
xy + CA y 2 ).
A
(b) Complete the square and write the equation in the form
2
B
B 2 − 4AC 2
z =A x+
y −
y .
2A
4A
(c) Explain why the surface is an elliptic paraboloid when B 2 − 4AC < 0, and a hyperbolic
paraboloid when B 2 − 4AC > 0.
6. (Take-home exam problem from 2012) There is a unique number A > 0 so that the circle
centered at (0, 0) with radius A intersects the line 3x + 2y = 13 exactly once. [ If r < A, the
circle of radius r doesn’t touch the line, and if r > A, the circle of radius r centered at (0, 0)
intersects the line twice. ]
Find the number A. Justify your answer with words and equations.