BC 3
A Quick Review of Vectors
1.
Name:
notation
v
initial point, terminal point
A
v
AB
B
2.
vector addition
parallelogram method
head to tail (consecutive) method
v  (v )  0 (the zero vector)
3.
scalar (real number) multiplication
v  v  2v
v  v  v  3v
v  1 v
k  v = (real number) · (vector) = vector,
4.
vector subtraction
v1  v2  v1  (v2 )
5.
distributive properties
 k1  k2  v  k1v  k2v ,
6.
k  v1  v2   k v1  kv2
vector components
2
Examples:
4v  4  3, 2  12, 8
3
If u  4, 1 , then v  u  3, 2  4, 1  1, 1
Example:
Find AB if the coordinates of points A and B are (3, –1) and (7, –3), respectively.
AB  OB  OA  7,  3  3,  1  4,  2
IMSA
Vectors 1.1
F14
7.
standard unit vectors: i  1, 0 and j  0, 1
All two-dimensional vectors may be written as a linear combination of i and j .
Examples:
8.
v  3, 2  3 i  2 j
a, b  a i  bj
a, b  a 2  b 2
magnitude of a vector = length of vector:
Example: | v | 3, 2  32  22  13
9.
unit vector  a vector of magnitude 1
1
unit vector in direction of u : uˆ 
u
| u|

1
3
2
 3, 2 
,
13
13 13
Note: This unit vector may be written as cos   i  sin   j for some angle .
Example: unit vector in direction of v  3, 2 : v 
10.
tangent vector to graph of a function (x) at x = a:
Find ( x) , then create the vector 1, (a) or x, (a)  x for some choice of x .
Example: Find a tangent vector to graph of   x   x 2  x at x  3 .
( x)  2 x  1   (3)  5  a tangent vector to (x) at x = 3 is 1, 5 .
11.
normal vector to graph of a function (x) at x = a:
Create a vector perpendicular to a tangent vector to graph of (x) at x = a:
Example: Find a normal vector to graph of   x   x 2  x at x  3 .
From #10, slope of perpendicular vector is 
12.
1
1
 normal vector is 1, 
or 5,  1 .
5
5
velocity and acceleration vectors
If R  t  is a vector-valued function such that R  t     t  , g  t  , then:

d R t 

dt
2
d R t 

dt 2
IMSA

   t  , g   t  is its velocity vector and
  t  , g   t  is its acceleration vector.
Vectors 1.2
F14