so now that we've seen a kind of precis
of vectors let's move on and talk about
vector operations before i begin
explicitly with vector operations let's
just detail for a moment why vectors are
useful in the context of building
mathematical models most often for
students vectors are commonly encounter
if not in a math class in a physics or
engineering class and just give you a
example of vectors and how they would
model real-world phenomena one such way
would be if let's say I have a mass just
in a physical sense minute you're
looking example here I've got a mass and
I can talk about let's say that mass is
resting on a surface a table or what
have you or the ground and I can talk
about the forces kind of acting on this
mass in a sense now in one sense of
course there always is the presence of
gravity so we can say the force due to
gravity which is of course pulling this
mass downward right toward the surface
of the earth now that force I can
describe and as a vector so I might call
this just F with an arrow over it using
our notation from just a moment ago and
the direction member of vector has
direction and magnitude and the
direction of that particular force due
to gravity would be going downward
toward the surface of the earth and I
could draw the length or the magnitude
of that vector to also express the
magnitude of the force and that's
typically described in units such as
Newton's and physics and whatnot also
let's make this picture a little more
interesting I could talk about maybe
there's another force over here let's
call this force will just denote the
vector V here and let's say there is a
maybe a person pulling the mass using a
rope and so there might be this force
displacing the mass to the right we
could then ask a question well what's
the total forces acting on the mass well
to answer a question like that we need
to impose some basic arithmetic alors
algebraic operations involving vectors
so what might it mean to add those
vectors together well that would be
asking the question what's the total
force being applied to this
Master this object so let's talk about
vector operations then on the heels of
that generally speaking there are four
prevalent vector operations in this
little sub unit here I'm going to talk
about scalar multiplication and vector
addition and then we'll get to the other
two vector operations in short order so
let's walk through these here with each
of these operations I'm going to explain
them and it's important to see this in
two lights in both the geometric and an
algebraic sense okay so the first such
operation is called scalar
multiplication and let's first talk
about well what a scalar is a scalar is
just for us here a number let's say a
real number so I'll denote that with a C
and of course we know a vector is a an
entity that has dimension so we'll say
our vector for instance lives in r2 so
with scalar multiplication what you do
is you operate on a scalar and a vector
together it's a binary operation and
it's typically written like this I'll
just give you the notation and just to
be complete let's say or vector consists
of once again components a and B in that
order so we denote scalar multiplication
by adjacency here so just in other words
right the scalar next to the vector I
don't put a dot or any multiplication
symbol between it typically and let's
sort of expand this notation so that's
saying C multiplied by the vector
described in component wise as a comma B
well the definition the algebraic
definition of scalar multiplication says
you just distribute the scalar across
each component of the vectors in other
words I'd have C times a comma C times
be there is the algebraic definition of
scalar multiplication let's do that with
the simple example that so for instance
if i have the scalar let's say two times
the vector 1 comma 3 once again i
distribute the 2 across the components
of the vector
pretty simple arithmetic here right so I
just go to x 1 is to 2 times 3 is 6
there is my new vector there is the
resultant vector of that scalar
multiplication now we should ask well
what does that mean geometrically in a
nutshell when you perform scalar
multiplication its scales as the name
would imply the vector so it's going to
change the magnitude but not the
direction importantly of the vector so
let's draw a picture of kind of what's
happening here so geometrically speaking
we'll say we start with our vector V
okay so I've just called this vector V
in our example 13 so meaning again the
weight along our x-axis is one the
weight or the component value along our
y-axis is three so I'll draw this vector
in standard position emanating from the
origin there is the vector V okay so
what does it mean to scale that vector x
2 well what happened here is we once
again maintain the direction of the
original vector but we've multiplied the
length of the magnitude of that vector
by tues I'll do my best to kind of draw
this to scale in a sense but here would
be now in blue the vector 2v nervous
it's twice the length of the original
vector but maintaining the same
direction in other words they're still
that angle theta that hasn't changed so
if I multiply by 2 I stretch the
original vector by a factor of 2 on the
other hand if I multiply the vector by
scalar value one half it would have the
length of the vector and still maintain
the same direction what happens if i
multiply a vector by let's say a
negative scalar value again we'll use
our same vector example well
algebraically i distribute that scalar
across the components and what do I have
here I have the exact same vector that
original vector now just pointing in the
opposite direction so once again if
there is my vector V negative V would
just reflect through the origin and
would point in the opposite direction
similarly if i multiplied by negative 2
it would flip the orientation of the
vector so it point in the opposite
direction but it would also scale it by
value 2 so this would be the vector
negative 2