Sec. 1.2
3,4,5,6,8,9,11,12,16,17,21,23
3.
v
u
U
v
uU
±
1
5
u
3,4
U
and
w
u
U
w
u U
±
1
10
u
8,6
U
give unit vectors in the directions of
v
u
,
w
u
Then
cos
u
±
v
u
±
w
u
U
v
uUU
w
u U
±
v
u
U
v
±
w
u
U
w
u U
±
1
5
u
U ±
1
10
u
U
±
48
50
±
24
25
,
and
u
±
cos
u
1
24
25
±
0.28379
(in radians)
Given
w
u ±
u
U
the vectors
u
4,3
U
,
u
u
U
,
u
u
4,
u
3
U
make an angle of
0
U
, 90
U
, 180
U
,
respectively, with
w
u
.
4. a)
v
u
± u
u
v
u
U
±
u
u
v
u
±
v
u
U
±
u
U
v
2
±
u
1
b) Expand, the crossterms cancel:
u
v
u ²
w
u
U ± u
v
u
u
w
u
U
±
v
u
±
v
u
u
w
u
±
w
u ±
0
(the diagonals of
a rhombus meet at right angles)
c)
u
v
u ²
2
w
u
U ± u
v
u
u
2
w
u
U
±
1
u
4
±
u
3
5.
ū
1
±
1
10
u
3,1
U
,
ū
2
±
1
3
u
2,1,2
U
Ū
1
±
u
u
1,3
U
and
Ū
2
±
u
1,
u
2,0
U
or
u
0,
u
2,1
U
are perpendicular to
ū
1
and
ū
2
respectively. Note
that any combination
c
u
1,
u
U
²
d
u
0,
u
U
is perpendicular to
ū
2
and that exhausts all the
possibilities.
6. a) Any multiple of
u
1,2
U
is perpendicular to
u
2,
u
1
U
.
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 Spring '08
 STAFF
 Linear Algebra, Algebra, Vectors, Cos

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