-
vector: a quantity having both direction and magnitude (length).
A two-dimensional vector is an ordered pair a
;
A three-dimensional vector is an ordered triple
a 
. The numbers 
, 
, and 
are called
components of a.
-
A representation of the vector a is a directed
line segment
from any point A(x,y) to the point
. A particular representation of a is the directed
line segment 
from the origin to the point 
, called the
position vector of the point . Representations in 3-d are
defined analogously.
Vectors will generally be written with an arrow, e.g.
, since
it's hard to write boldface on paper or on the board.
-
parallel vectors: if two vectors are parallel, then one can be written
as a scalar multiple of the other:
- unit vector: a vector whose length is 1.
None to speak of.
-
Given the points
and 
, the vector a
with representation is
-
The length of the two-dimensional vector a is
The length of the three-dimensional vector a
is
-
Addition is defined componentwise, so that a + b is
defined as
Subtraction is defined in the obvious way.
Addition is carried out geometrically by putting the tail of vector b to the head of a and creating the vector from the tail of a to the head of b, creating a parallelogram.
-
Multiplication of a vector by a scalar: If c is a scalar (e.g., a real
number) and a , then the vector ca is defined
by
and similarly for three-dimensional vectors.
-
Properties of vectors:
-
Special vectors (the standard basis vectors, of length 1):
Any vector can be expressed as a sum of the standard unit vectors:
- Vectors can be defined in n-dimensions in entirely analogous ways.
This section simply introduces us to a quantity, called a vector, which allows us to capture both magnitude and direction. This is useful (for example to indicate wind speed and direction on a weather map), and a set of rules and properties are defined to help us to manipulate these quantities.
Every vector can be expressed as a sum of special ``basis'' vectors, which are of unit size (length 1)..