let's begin with the general notion of
what a vector is and talk about some
typical notation for vectors so to write
or denote a vector will write it let's
call that vector V for instance will
denote a vector with an arrow over it as
is typical and what I'm showing you here
is something called bracket notation so
we can write a vector and list the
elements of that vector in order so this
would be a two-dimensional vector and
the elements listed between the brackets
are known as components or the
components of that vector so because
this is a two-dimensional vector we can
use our previous notation and write that
that vector is an element of our two
meaning those components are real
numbers for instance okay these are just
placeholders with a and B similarly I
could define a three-dimensional vector
also in bracket notation I'll just label
the components a b and c respectively
and because this is a three-dimensional
vector we would then therefore say that
this vector lives in r3 so that is the
basic underpinning of an algebraic
notation for vectors I'd like to also
make this notation cohere with a
geometric interpretation which is common
for vectors so how do I understand a
vector and a geometric sense will have
vectored it's an object mathematical
object that has both a direction and a
magnitude ok so here's our basic
geometric interpretation of vector is in
let's say an arrow with a direction and
magnitude the defining features of a
vector so what's meant by direction and
magnitude well the direction you can
think of kind of an angle defined here
with the x-axis will call that theta as
it's typical with vectors you can think
of this vector V that I've drawn in the
plane here's my x-axis my y-axis of
course this Victor has an angle
associated with it that defines its
directions and the magnitude of that
vector is just the length of the vector
so there we have the algebraic notion of
a vector we can write it in bracket
notation by simply listing
components in order of that vector we
can also realize a vector in a geometric
senses an arrow in a plane or in 3-space
I can also draw that as well with a
direction and the magnitude so will this
look like in three space well we just
kind of extend things in an intuitive
way geometrically so I have an X X and Y
axis I need three axes of course here to
define three space this would be just an
image of our three if I want to define a
vector in 3-space let's see let's say
for instance we're going to define the
vector 1 2 3 in our three so that means
I move in the X direction with weight 1
in the Y direction with weight too and
in the Z direction with wait 3 so again
it's a little bit hard to draw in three
space but you can imagine here's kind of
the shadow of that point and there is my
vector in r3 so it's it's arrow
emanating from the origin and we have a
vector because it has a direction we
typically specify where it begins so
this would be called the initial point
of the vector in this case this is a
vector with initial point at the origin
and this point where it ends is usually
referred to as the terminal point of the
vector ok so we start the initial point
of course and when you go in the
direction of the terminal point that
defines the direction of the vector one
more little piece of notation when a
vector initiates at the origin or its
initial point is at the origin that
vector is said to be in standard
position and one other sort of nice
aspect of vectors that makes kind of our
math max consistent is I can take any
vector let's say this vector V and I've
just sort of done my best here to to
create a carbon copy of that vector V I
can move that vector anywhere I want in
the space let's say in the plane for
instance and even though it doesn't have
the same initial point and end point it
does nonetheless have the same direction
let's say there's the same angle theta
and the same magnitude I you the same
length so when i create a carbon copy of
a vector displace it somewhere else in
space these two vectors are called
equivalent vectors in other words
they're mathematically identical and a
nice aspect of vectors is when i do
computations and i do
involving vectors which we'll see in
just a moment it doesn't matter where I
place the vector in other words what i
call the origin the mathematics will be
consistent throughout so we're going to
get similar results for equivalent
vectors okay so just in summary a vector
basically for our purposes here is an
arrow with the direction and a magnitude
I can express the vector with an arrow
over it to denote that it has dimension
and the in algebraic sense I just list
the components of that vector in order
so a B this is a two-dimensional vector
this is a three dimensional vector it
lives in r3 and I can similarly draw a
vector in r2 and r3 to capture that same
idea