MATH 2203 –Exam 1 (Version 1) Solutions
February 11, 2008
S. F. Ellermeyer Name
Instructions. Your work on this exam will be graded according to two criteria: mathe-
matical correctness and clarity of presentation. In other words, you must know what
you are doing (mathematically) and you must also express yourself clearly. In particular,
write answers to questions using correct notation and using complete sentences where
appropriate. Also, you must supply su¢ cient detail in your solutions (relevant calculations,
written explanations of why you are doing these calculations, etc.). It is not su¢ cient to just
write down an “answer” with no explanation of how you arrived at that answer. As a rule
of thumb, the harder that I have to work to interpret what you are trying to say, the less
credit you will get. You may use your calculator but you may not use any books or notes.
1. Describe in words (writing a complete sentence or sentences) the region of <3repre-
sented by the inequality 4< x2+y2+z2<9.
Answer: This region consists of all points between (but not on) the spheres of radius
2 and 3 centered at the origin.
2. Let aand bbe the vectors a=h2;1iand b=h5;7i.
(a) Compute jaj,jbj, and ab.
(b) Find the angle between aand b(to the nearest tenth of a degree).
Solution:
jaj=q(2)2+ (1)2=p5
jbj=p52+ 72=p74
ab= (2) (5) + (1) (7) = 17.
Also, if is the angle between aand bthen
cos () = ab
jajjbj=17
p5p74
so
= arccos 17
p5p74152:1.
3. Find a unit vector that is orthogonal to both i+jand i+k.
Solution:
(i+j)(i+k) = ik+ji+jk
=jk+i
=ijk
is orthogonal to both i+jand i+k. Since the vector we found had magnitude p3, a
unit vector that is orthogonal to both i+jand i+kis
u=p3
3(ijk).
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