1.
2.
notation
initial point, terminal point
r
v
A
r
v
B
vector addition
parallelogram method
head to tail (consecutive) method
r
v
r
u
r r
uv
r r
uv
r
u
r
v
3.
uuur
AB
r
r r
v  (v)  0 (the zero vector)
scalar (real number) multiplication
r r r
r
r
r
r r
r
v  v  v  3v
v  1  v
v  v  2v
r
k  v = (real number) · (vector) = another vector,
4.
vector subtraction
ur
v1
ur ur ur
ur
v1  v2  v1  (v2 )
ur uur
v1  v2
uur
v2
5.
distributive properties
r
r

6.

ur ur
ur
ur
k v1  v2  k v1  k v2
r
 k1  k2  v  k1 v  k2 v ,
vector components
r
v  3, 2
2
Examples:
r
4v  4  3, 2  12, 8
r
r r
If u  4, 1 , then v  u  3, 2  4, 1  1, 1
3
Example:
uuur
Find AB if the coordinates of points A and B are (3, –1) and (7, –3), respectively.
uuur uuur uur
AB  OB  OA  7,  3  3,  1  4,  2
7.
r
r
r r
r r
r r
dot product: If u  a, b and v  c, d , then u gv  ac  bd ; u gv  0 if and only if u ,v are parallel.
ur
8.
ur
standard unit vectors: i  1, 0 and j  0, 1
ur
ur
All two-dimensional vectors may be written as a linear combination of i and j .
9.
ur
r
a, b  a i  b j
r
ur
r
v  3, 2  3 i  2 j
Examples:
a, b  a 2  b2
magnitude of a vector = length of vector:
r
| |
Example: v  3, 2  3  2  13
10.
2
2
unit vector  a vector of magnitude 1
r

1
r
unit vector in direction of u : u  r  u
|u|

r
1
Example: unit vector in direction of v  3, 2 : v 
 3, 2 
3
,
2
13
13 13
ur
ur
Note: This unit vector may be written as cos  i  sin   j for some angle .
11.
tangent vector to graph of a function f  x  at x = a:
Find f   x  , then create the vector 1, f   a  or x, f   a   x for some choice of x .
Example: Find a tangent vector to graph of f  x   x  x at x  3 .
2
f   x   2x  1
12.
f   3  5  a tangent vector to f  x  at x = 3 is 1, 5 .
normal vector to graph of a function f  x  at x = a:
Create a vector perpendicular to a tangent vector to graph of f  x  at x = a:
Example: Find a normal vector to graph of f  x   x  x at x  3 .
2
From #10, slope of perpendicular vector is 
13.
1
1
or 5,  1 .
 normal vector is 1, 
5
5
velocity and acceleration vectors
uur
uur
If R  t  is a vector-valued function such that R  t   f  t  , g  t  , then:

uur
d R t 

dt
uur
2
d R t 

dt
2

f   t  , g   t  is its velocity vector and
f   t  , g   t  is it’s a acceleration vector.