Boian Popunkiov / Math 234 17 January 2005
Wisconsin Emerging Scholars Worksheet 1
Problem 1 If ~u,~v, and ~w are vectors and all the other quantities are scalars which of the following
expressions make sense
1. ~u ·(~v ×~w)
2. (~u ·~v)×~w
3. ~u ×~v +k
4. ~u ·~v +k
5. (~u ×~v)×~w
6. (~u ·~v)·~v
7. (~u ·~v)~w
Problem 2 Show that for any two vectors ~u and ~v we have
|~u +~v|2+|~u −~v|2= 2¡|~u|2+|~v|2¢.
What geometric theorem about a parallelogram can you deduce from that?
Problem 3* Three vectors ~u,~v, and ~w. satisfy all the following properties:
|~u|=|~w|= 5,|~v |= 1,|~u −~v +~w|=|~u +~v +~w|.
If the angle between ~u and ~v is π/8, what is the angle between ~v and ~w.
Problem 4* Either show that the statements are true or give a counterexample. In each case ~w 6=~
0.
(a) If ~u ·~w =~v ·~w, then ~u =~v .
(b) If ~u ×~w =~v ×~w, then ~u =~v .
(c) If If ~u ·~w =~v ·~w and ~u ×~w =~v ×~w, then ~u =~v .
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