Math 208 Worksheet 5-4-11
Projection: the aim here is to find the component of a in the direction of b. It can be used to compute
work using a = force vector, b = displacement vector. The scalar projection is just the signed
magnitude of the projection.
1)
2)
3)
4)
5)
Compute the scalar and vector projections of <1, 3, 2> onto <0,3,1>
Compute the scalar and vector projections of <0,-3,-4> onto <1,2,4>
Find the angle between <1, 3, 0> and <-3, 2, 1>
Find b so that <-1, b, 2> and <b, b, -1> are orthogonal.
Determine whether the series converges conditionally, absolutely or divergent:
cos n

2
n 1 n  2n

a)
( 1) n

b)
 tan n 
1
n 1
n
6) (10.3 #42 HW problem) Suppose all sides of a quadrilateral are equal in length and opposite sides
are parallel. Use vectors to show that the diagonals are perpendicular.
(hint: use the picture shown below: we may place one of the vertex at the origin. Let the length
be a. Also let A=(0,0), B=(c,d).
a) First find the x and y components of C and D.
b) Then compute the diagonal vectors AD and BC
c) Compute their dot product
d) Find a relationship using a, c, and d.
proj 
1)
a
Vector :
2)
ab
2
a
 1,3,2  0,3,1 
 0,3,1 
2
 0,3,1  0,
33 11
, 
10 10
a  b  1,3,2  0,3,1  11


a
0  9 1
10
ab
 0,3,4  1,2,4 
12 64
proj  2 a 
 1,2,4  0, ,

2
21 21
a
 1,2,4 
: Scalar:
Scalar:
a  b  0,3,4  1,2,4   10


a
1  4  16
21
3)
 ab 
  cos 1   3 

 10 14 
 ab 
  cos 1 
4) Choose b so that their dot product is 0.
 b  b2  2  0  b  2,1
5) A) Absolute convergent by the comparison test since
B) Absolute convergent by the comparison test since
lim tan 1 n 

n
2
 1.57 so it passes 1.5 after while.
1
1
1
 2
 2
2
n  2n n  2n
3n
3
tan 1 n  for large n since ,
2
Thus
2
1
tan n
1
n

2
 
n
3
3
 
2
1
n
,a
convergent geometric series.
6) Since the length is a, D=(a,0), C=(c+a, d). Then AD=<c+a, d> and BC=<a-c, -d>. Their dot
product is a  c
perpendicular.
2
2
 d2.
But this is 0 by the Pythagoran Theorem. Thus the diagonals are