MATHEMATICS 226, FALL 2014, PROBLEM SET 11
Due on Friday, September 12
Write clearly and legibly, in complete sentences. You must provide complete explanations for all your solutions; answers without justification, even
if correct, will not be marked. You may discuss the homework with other
students, but the final write-up must be your own.
1. (10 marks) Specify the boundary and the interior of the sets S in 3space whose points (x, y, z) satisfy the given conditions. Is S open,
closed, or neither?
(a) x2 + y 2 + z 2 ≥ 16
(b) z ≥ 0, x2 + (y − 2)2 + z 2 < 1
2. (10 marks) Find all values of t for which the vector v = 4ti − tj + 6k
is perpendicular to the vector w = 2i + (4 − 2t)j + (1 − 2t)k.
3. (10 marks) Find vectors a, b, c in R2 such that the set of points whose
position vector r satisfies the inequalities
r · a ≤ 1, r · b ≤ 1, r · c ≤ 1
is the triangle with vertices (1, 2), (2, −2), (−3, 0).
4. (10 marks) Find two unit vectors each of which makes equal angles
with the vectors u = 4i − j − k, v = i + j, and w = 2i + j + k.
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