MATH 20C Lecture 2  Monday, September 27, 2010
Observations
1. Two vectors pointing in the same direction are scalar multiples of each other.
2. The sum of three headtotail vectors in a triangle is 0
.
Lines in space
Polled the students about what kind of object is described by the equation
x
+ 2
y
= 5 in the plane.
Most got it right (a line). How about the equation
x
+ 2
y
+ 3
z
= 7 in space? It’s a plane (about
half of the students answered correctly, the other half thought it was a line).
So need some other way to think about lines in space. Think of it as the trajectory of a moving
point. Express lines by a parametric equations.
In coordinates.
For instance, the line through
Q
0
= (1
,
2
,
2) and
Q
1
= (1
,
3
,
1): moving point
Q
(
t
) = (
x, y, z
) starts at
Q
0
at
t
= 0, moves at constant speed along line, reaches
Q
1
at
t
= 1. Its
“velocity” is
~v
=
→
Q
0
Q
1
, so
→
Q
0
Q
(
t
) =
t
→
Q
0
Q
1
.
In our example we get
h
x
+1
, y

2
, z

2
i
=
t
h
2
,
1
,

3
i
,
i.e.
x
(
t
)
=

1 + 2
t
y
(
t
)
=
2 +
t
z
(
t
)
=
2

3
t
In vector form.
The line through a point
P
in the direction given by some vector
~v
is given by
~
r
(
t
) =
→
OP
+
t~v
where
~
r
(
t
) is the position vector of the point
Q
(
t
) on the line. Did example with
P
= (0
,
1
,
1) and
v
=
h
0
,
5
,

1
i
.
(Not sure I remember the exact numbers, I had made it up on the spot in class.)
Dot product
Definition
~
A
·
~
B
=
a
1
b
1
+
a
2
b
2
+
a
3
b
3
+
. . .
(a scalar, not a vector)
Geometrically
~
A
·
~
B
=

~
A

~
B

cos
θ,
where
θ
is the angle between the two vectors.
Explained the result as follows. First,
~
A
·
~
A
=

~
A

2
cos 0 =

~
A

2
is consistent with the definition.
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 Spring '08
 Helton
 Vectors, Scalar, Vector Space, Dot Product

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