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Quiz #1
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#1.(3 pt’s) Let u  (3, 4,1) and v  (5, 2,1) . Calculate projv (u ) .
6
 u v 
 2 1 
projv u  
(5, 2,1)  1, ,  
v 
30
5
 v v 
 5
#2(3 pt's) Find the parametric equations of the line through (1, 4, 2) and (5, 4, 2) .
 x  1  4t

M  (4, 0, 4) is parallel to the line. So the parametric equations are:  y  4
 z  2  4t

#3(4 pts) Determine if the lines given by the parametric formulas (2, 2, 3) + t(1, 0, -2) and
(6, 3, 1) + s(4, 2, 4) intersect. If so, at what point do they intersect?
We need:
 2  t  6  4s
 1 
 ( s, t )   , 2 

 2 
2  0t  3  2s
 1 
This gives ( x, y, z )  (4, 2, 1) . Note that ( s, t )   , 2  gives this same point in both lines.
 2 
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#4(3 pt’s) Prove that for u , v 
Proof: Let u , v 
3
3
: u v  u v
. Then by definition, u  v  u v cos  = u v cos  . Since 1  cos   1,
it follows that cos   1.
Therefore, u  v  u v cos  u v .
Q.E.D.
Note: equality holds if and only is   0 or 180
#5(3 pt’s) Find three non-parallel non-zero vectors orthogonal to u  (3, 4,1).
Many answers here: e.g. (4,3, 0), (1, 0,3), and (0, 1, 4) . Note that the dot product of any of
these vectors with u is 0.
#6(5 pt's) Let x  (2, a,1  a) and y  (6, 9, 6)
a. Find all values of a such that x and y orthogonal.
x and y orthogonal  x  y  0  12  9a  6(1  a)  0  a 
b. Find all values of a such that x and y parallel.
Looking at the first components
x and y parallel  y  3x  3a  9 and  3(1  a)  6  a  3
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2
5