Download Given the following vectors u and v, compute the things listed in

Problem 1.
problem.
[8 points] Given the following vectors u and v, compute the things listed in each part of this


2
u= 3 
1


1
v =  −1 
2
(a) The dot product u · v.
Solution. u · v = 2 · 1 + 3 · (−1) + 1 · 2 = 1.
(b) The distance between u and v.
Solution. The distance is ku − vk; since


1
u−v = 4 
−1
we hare ku − vk =
√
18.
(c) A unit vector in the same direction as u.
Solution. This is given by
1
kuk u;
we can compute kuk =
√
14 and thus the vector we want is
√

2/√14
 3/ 14  .
√
1/ 14

Problem 2. [12 points] Let W be the subspace determined by the vectors w1 , w2 below (which are orthogonal to each other). Decompose the vector y given as a sum yb + z for a vector yb in W and a vector z
perpendicular to W .






1
1
1
 2 
 −1 
 1 



y=
w2 = 
w1 = 
 4 
 −1 
 1 
1
1
1
Solution. The formula for yb is

yb =
The vector z is then
y · w1
w1 · w1
w1 +
y · w2
w2 · w2


1

8 1 
 + −4 
w2 = 


4 1
4 
1


0
 −1 

z = y − yb = 
 1 .
0
1
 
1
1

−1 
= 3
−1   3
1
1


.
