Vectors
Outline of Hass, Weir, Thomas –
Section 9.2
Definition
A vector is a directed line segment.
segment The directed
line segment AB has initial point A and teminal
point B.
B Its length is denoted by AB . Two vectors
are equal if they have the same length and direction.
Definition
If v is a two–dimensional vector equal to the vector
with initial point at the origin and terminal point v
 1 , v 2 ,

then the component form of v is
v  v 1 , v 2 .
If v is a three–dimensional vector equal to the vector
with initial point at the origin and terminal point v 1 , v 2 , v 3 ,
then the component form of v is
v  v 1 , v 2 , v 3 .
Definition
If Px 1 , y 1 , z 1  and Qx 2 , y 2 , z 2  are points in R 3 , then we
define the magnitude or length of the vector v  PQ to
be the distance between the points P and Q. The length
of v is denoted by |v|.
|v| Thus
|v|  PQ 
x 2  x 1  2  y 2  y 1  2  z 2  z 1  2 .
The only vector with length 0 is the zero vector,
0  0,
0 0,
0 0 .
This is also the only vector with no specific direction.
Example 1
Find the component form and the length of the vector
with initial point P1, 2, 3 and terminal point Q3, 2, 2.
Example 2
A small cart is being pulled along a smooth horizontal
floor with a 20 lb force, F, making a 45  angle to the
floor (Figure 9.11). What is the effective force moving
the cart forward?
Definitions
Let u  u 1 , u 2 , u 3  and v  v 1 , v 2 , v 3  be vectors and let
k be a scalar. We define vector addition and multiplication
of a vector by a scalar as follows:
u  v  u 1  v 1 , u 2  v 2 , u 3  v 3 
ku  ku 1 , ku 2 , ku 3 .
Also, the vector u is defined to be 1u and the difference
u  v is defined to be u  v.
See figures 9.12, 13, and 14 for geometric illustrations of
th
these
concepts.
t
Important facts about scalar
multiplication
l l
If k  0,
0 then the vector ku has the same direction as
the vector u. If k  0, then the vector ku has the opposite
direction of the vector u. In both cases,, the length
g of ku
is |k| times the length of u. In other words,
|ku|  |k||u|.
Example 3
Let u  2, 1, 1  and v  5, 5, 2 . Find
(a) 2u  3v
(b) u  v
( ) 12 u .
(c)
Properties of Vector Operations
Let u, v, and w be vectors and let a and b be scalars.
1) u  v  v  u
2) u  v  w  u  v  w
3) u  0  u
4) u  u  0
5) 0u  0
6) 1u  u
7) abu  abu
8) au  v  au  av
9) a  bu  au  bu
Example 4
A jetliner flying due east at 500 mph in still air
encounters a 70 mph tailwind blowing in the
direction 60  north of east. The airplane holds
its compass heading due east but, because of
the wind, acquires a new ground speed and
direction. What are they?
(Refer to Figure
Fig re 9.15.)
9 15 )
Unit Vectors
A vector of length 1 is called a unit vector. The
basic unit vectors are
i  1, 0, 0
j  0,
1 0
0 1,
k  0, 0, 1 .
Any vector, v  v 1 , v 2 , v 3 , can be written as a
linear combination of i, j, and k. In particular,
v  v 1 i  v 2 j  v 3 k.
The Direction of a Vector
If v  0,
0 then
th |v|
0 Also,
Al
th vector
the
t
| |  0.
1 v
|v|
has length 1 and points in the same direction as v.
Thus
1 v
|v|
is a unit vector that points in the same direction as v.
We call this vector the direction of v.
Example 5
Find the direction of the vector v  3, 1, 2 .
Example 6
The velocity of a moving object is a vector, v,
that tells us how fast something is moving (the
speed) and in what direction the thing is moving.
The speed is the magnitude of v.
v
Suppose that a moving object has velocity vector
v  2i  4j.
Find the speed and direction of the motion.
Example 7
A force of 6 newtons is applied in the direction of
the vector v  2i  2j  k. Express the force, F, as
a product of its magnitude and direction.
Midpoint of a Line Segment
The midpoint,
p
, M,, of the line segment
g
jjoining
g the
points P 1 x 1 , y 1 , z 1  and P 2 x 2 , y 2 , z 2  is
M
x 1 x 2
2
,
y 1 y 2
2
,
z 1 z 2
2
.
Refer to Figure 9.17 for the proof (which will be done in class).