in this video we introduced the idea of the opposite of a vector both geometrically and
algebraically next we introduced the zero vector as the result of adding a vector to its opposite
and finally we defined vector subtraction as the addition of the opposite and we learn to
quickly visualize vector differences in a geometric way so we start by defining the
opposite of a vector v as a vector with the same magnitude but the opposite direction
and we call that negative v next we imagine adding a vector to its opposite so there's a vector v and
to vector add this to the opposite we have to take the tail of the opposite vector negative v
and attach it to the head of v itself and then we draw the vector from the tail of the first vector
to the head of the second but the head and tail are in the same place the result is a vector of
length zero and that's called the zero vector don't confuse the zero vector with the scalar
zero in order for the equation v vector plus the opposite of v equals zero to make sense the
data type has to be the same on both sides of the equation those are vectors on the left that's got
to be a vector on the right so that zero vector actually has two coordinates and they're both
zero next we have a little note about scale or multiplication by negative 1. so given the vector
v as a row vector 3 5 i want to show graphically that negative 1 times v is the opposite of v so
it's useful to write this in the i-hat j-hat notation v is 3 i hat plus 5 j hat so what's
negative 1 times v just distribute the negative 1 to both terms and i get negative 3 i hat minus 5 j
hat let's take a look at these vectors graphically there's v it has an x component of 3i hat and
a y component of 5 j hat and there's negative 1 times v with this x component of negative 3 i hat
and its y component of negative 5 j hat and i can see that this produces a vector with the
same magnitude and exactly the opposite direction in other words negative one times v is negative v
the thing that we call the opposite finally i want to show algebraically that if i add v to negative
one times v i end up with zero so there's v and i add it to negative one times v and all i have to
do to add these vectors is just collect the like terms and i have a 3i hat and a negative 3i hat
that gives me 0 i hat and i have a 5 j hat and a negative 5 j hat and that gives me zero
j hat and again i'm taking care to write this as a vector it's the zero vector it has two
components that are both zero now we can use the idea of the opposite to define vector subtraction
so what do you do to find the difference between two vectors v minus w you add the opposite this
is exactly the same thing we do with scalars if i told you to find the difference five minus three i
could say that i start with five and i add the opposite of three and of course i get two from
that calculation well now we can do the same thing with vectors if i want v minus w i think of it as
v added to the opposite of w and we have a good definition for the opposite of w now so we can
do this geometrically the opposite of w is a vector of the same length as w but pointing in
the opposite direction and i'm trying to vector add that to the vector v so i have to attach the
tail of it to the head of v and then i draw the vector from the tail of v to the head of negative
w i can also do vector subtraction algebraically so in the next example i have v is 2 i hat minus j
hat w is negative 2 i hat plus 3 j hat and i want to get v minus w so i just plug in v is 2 i hat
minus j hat and then subtract negative 2 i hat plus 3 j hat alternatively you could say you're
adding negative 1 times that quantity we have to distribute that factor of negative 1. i have
a double negative on that i hat component then we collect like terms this gives me four i hat
minus four j hat and that's just an algebraic way of doing the same thing we did geometrically
in the previous example finally there's a very useful construction to remember about vector
differences and we use this all the time in physics in this last picture i have vectors
v and w with their tails located in the same place and if i do the same sort of construction
that i did on the left i take the opposite of w and then i draw a vector from the tail of v to
the head of that negative w and that gives me v minus w okay so there's nothing wrong with that
but the powerful new fact that i'm after here is that that vector difference i just drew is exactly
the same as drawing a vector from the head of w to the head of v that also gives you v minus w
so when you have two vectors located with their tails in the same place and you take
their difference it creates a vector pointing from the head of one to the head of the other
and i can verify that this makes sense by seeing that w and the difference v minus w are being
vector added here because the tail of the second is stuck to the head of the first and when i take
w plus v minus w the w's cancel and i end up with a vector sum of v which is exactly what i see in
the picture here so i like to use that little argument to remind myself which way it works
it's really going to pay off if you can remember that v minus w provided their tails are in the
same place is a vector pointing from the head of w to the head of v if you enjoyed this video
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