### Video Transcript

In this video, weโre gonna look at
what happens when you add or subtract vectors to or from each other. We know that vectors can be
represented by line segments with a specific length and direction, and using these
will really help us to visualise vector addition and subtraction.

The addition process amounts to
laying vectors end-to-end and calculating the components of the resultant vector by
examining the difference between the coordinates of the initial point of the first
vector and the terminal point of the last vector in the chain. So letโs see an example of
that.

Weโve got ๐ which is the
vector three, two and ๐ which is the vector four, negative one. And we want to add those two
vectors together. So just quickly sketching those
out, here weโve got vector ๐ which has an ๐ฅ-component of positive three and a
๐ฆ-component of positive two โ so weโve gone three in that direction, two in
that direction โ and ๐, which has an ๐ฅ-component of four and a ๐ฆ-component of
negative one.

So if we add these two vectors,
itโs like laying them end to end and then working out how we get from the
beginning of the first vector to the end of the last vector. So to get from here to here,
weโve taken a journey of three in the ๐ฅ-direction here and another four in the
๐ฅ-direction here. So to find the ๐ฅ-component of
the resultant vector, ๐ plus ๐, we simply add the ๐ฅ-component of ๐ to the
๐ฅ-component of ๐.

And now letโs visit the
๐ฆ-components. So to get from here to here
again, weโve done our positive two to take us up to the top of the diagram but
then we came back one. So weโve got to add those two
components together. So thatโs two plus negative
one. And three and four makes seven,
so the resultant ๐ฅ-component is seven. And two add negative one is
one, so the resultant ๐ฆ-component is one.

So addition of vectors is just
a case of adding the ๐ฅ-components and adding the ๐ฆ-components.

Now letโs add three vectors
together.

So weโve got vector ๐, which
is four minus two; ๐ is one, six; and ๐ is negative five, negative four. And weโve got to add all three
of those vectors together.

So first letโs just sketch out
a diagram. Well hereโs vector ๐ going
positive four in the ๐ฅ-direction and negative two in the ๐ฆ-direction. Thereโs ๐ going positive one
in the ๐ฅ-direction and positive six in the ๐ฆ-direction. And lastly we lay ๐ onto the
end of ๐, and weโve got negative five in the ๐ฅ-direction and negative four in
the ๐ฆ-direction. And we can see that when weโve
added ๐, ๐, and ๐, the initial point of the first vector ๐ is in exactly the
same place as the terminal point of the last vector ๐.

So although weโve gone around
the houses, we started off here and weโve ended up in exactly the same place
here. Okay letโs add those
๐ฅ-components together. So weโve gone four and then
another one, and then weโve taken away five. And in the ๐ฆ-direction, we
went negative two then added six and then finally took away four. And four add one take away five
is zero, and negative two add six take away four is also zero.

So the resultant vector by
doing the adding up of the components is zero, zero. And that tallies with what we
knew at the beginning: that we started off and we ended in exactly the same
position. So the resultant vector, we
havenโt moved in the ๐ฅ-direction we havenโt moved in the ๐ฆ-direction; our
initial point and our terminal point are coincident.

Having looked at some basic
addition of vectors, letโs take a quick look at repeated addition of vectors, or as
itโs more properly called multiplying vectors by scalars. So letโs start up with ๐ is equal
to the vector two, three. Letโs say we wanted to find out the
what the value of two ๐ was. Well two ๐ is just ๐ plus ๐. And as weโve just seen, to add two
vectors together you just add the components together, so two plus two and three
plus three.

So yep weโre labouring the point
here a little bit. But that means, weโve got two lots
of the ๐ฅ-component and two lots of the ๐ฆ-component, so this two here. Weโve multiplied each of the
components by two, which gives us the vector four, six. Okay letโs think about three ๐
now, so thatโs ๐ plus ๐ plus ๐.

Well Iโm just gonna jump straight
to the stage. When weโre multiplying the
components, because itโs three ๐ weโre multiplying the ๐ฅ-component by three and
the ๐ฆ-component by three, which gives us six, nine. And letโs just look at one more
example: a fractional scalar, so a quarter. So a quarter of ๐, weโre just
gonna multiply each component by a quarter. So thatโs a quarter of two and a
quarter of three. So that gives us a half and
three-quarters as our components.

So just to summarise that, the
process of multiplying a vector by a scalar, we take the scalar and we just multiply
each component of the vector by that scalar to get our components of the resultant
vector. So letโs start off with the same
vector ๐. And now letโs have a look at
negative scalar multiplication, works in exactly the same way. So letโs just see a couple of
examples.

Okay weโre trying to find negative
๐. That just means minus one times ๐,
so weโre gonna take the scalar negative one and weโre gonna multiply both components
by negative one. And that gives us an answer of
negative two and negative three. And okay, letโs look at negative
two ๐. So this time the scalar is negative
two and weโre gonna multiply each of our ๐ฅ- and ๐ฆ-components by negative two. And that gives us an answer of
negative four, negative six.

Now itโs just worth taking a moment
to notice ๐ and negative ๐: the components have the same numbers, two- two and two
for the ๐ฅ-components and three and three for the ๐ฆ-components. But the in the negative ๐, the
signs are the opposite signs. In fact, what weโre doing is weโre
going exactly the opposite direction. So vector ๐, weโre going positive
two in the ๐ฅ-direction and positive three in the ๐ฆ-direction; thatโs this journey
here. Negative ๐, weโre going negative
two in the ๐ฅ-direction and negative three in the ๐ฆ-direction.

So weโve done the exact reverse
journey in the neg- in the negative direction, in the opposite direction. So positive ๐ and negative ๐ are
the same length of line, the same angle of vector, but theyโre just going in the
opposite direction.

Okay letโs look at a quick example
or two of subtracting some vectors.

So weโve got vector ๐ is two,
three and vector ๐ is four, negative one. And we wanna calculate the
resultant of vector ๐ take away vector ๐. Well thereโs different ways,
you could just take the ๐ฅ-components of ๐ and subtract the ๐ฅ-components of ๐
and the ๐ฆ-components of ๐ subtract the ๐ฆ-component of ๐ to get your
resultant. But I just like to suggest that
we look at it in this slightly different way so ๐ take away ๐ is the same as
๐ add the negative of ๐.

So when we write that out,
weโve got two plus the negative of four and we got three plus the negative of
negative one, which looks a bit on odd on the page. But there it is. So two add negative four is
negative two, and three add the negative of negative one, which is positive one,
makes four.

Now this business of converting
๐ minus ๐ into ๐ plus the negative of ๐ seems like weโre overcomplicated
things. But letโs just take a look at
it visually first and see how that pans out. So weโve got here vector ๐,
which Iโm going across two and up three. So if I was adding ๐, Iโd be
going right four and down one, but Iโm adding the negative of ๐. So adding the negative of ๐
takes me from the end of ๐ left four, so weโre going four in that
direction. And the negative of negative
one, weโre going up one in that direction.

So this idea of laying vectors
end to end, thinking of taking away ๐ as adding negative ๐, means we can just
lay down the negative vector to the end of ๐ and we can then look for this
resultant vector, going from the beginning of ๐ to the end of ๐ here. And when we do that, we were
going negative two in this direction and up here positive four in that
direction. So when subtracting vectors, we
can just subtract the components if you want to or we can add the negative of
the components for the second vector, which helps us to visualise the vector
diagram a bit more easily.

Letโs just take a look at one more
example. So weโre gonna start off with the
same ๐ and ๐ vectors.

But this time weโre gonna
calculate ๐ take away ๐, which is the same as ๐ add the negative of ๐, which
gives us four add negative two and negative one add negative three. So thatโs a resultant vector of
two, negative four.

Well letโs look at the diagram
again. Well first of all, vector ๐,
weโre going positive four in the ๐ฅ-direction and negative one in the
๐ฆ-direction and then adding the negative of ๐, thatโs negative two in the
๐ฅ-direction, so weโre coming back, negative two. And negative of three so weโre
gonna go down three โ one, two, three โ in the ๐ฆ-direction.

So we start off at the initial
point of ๐; we add the negative of ๐; and we end up at the terminal point of
๐. So our resulting journey is
this one here. And in completing that journey,
we went positive four in the ๐ฅ-direction but then we came back two, which gave
us an ๐ฅ-component of two. So this distance here is
two. And we went down one and then
we went down another three, which gave us our ๐ฆ-component of negative four,
which means weโve come down here four. So this vector here is ๐ take
away ๐, ๐ subtract ๐, or ๐ plus negative ๐.

So just putting those two
results side by side, ๐ take away ๐ was negative two four, and ๐ take away ๐
was two negative four. So they were both in the same
position here, this was the position. And this was the length of the
vector, but one was going in this direction and the other was going in that
direction.

So theyโre the same vector, but
in opposite directions; one vector is the negative of the other. So vector ๐ minus ๐ is the
negative of vector ๐ minus ๐. And that makes sense because
negative of ๐ is negative ๐ and the negative of negative ๐ is positive
๐. So algebraically, they mean the
same thing; and in vector format, they mean the same thing as well.

Now letโs look at one last
problem.

Weโve got a regular hexagon
๐ด๐ต๐ถ๐ท๐ธ๐น and ๐บ is the midpoint of that and we have to express ๐ด๐ธ in terms
of vectors ๐ข and ๐ฃ. So the vector ๐ฃ is from ๐บ to
๐ถ and the vector ๐ข is from ๐ท to ๐ถ. Now because this is a regular
hexagon, we know that a number of these things are parallel. So ๐ด๐ต and ๐ธ๐ท and ๐น๐บ and
๐บ๐ถ are all parallel; ๐ด๐น and ๐ต๐บ and ๐บ๐ธ and ๐ถ๐ท are all parallel; and
๐ธ๐น, ๐ท๐บ, ๐บ๐ด, and ๐ถ๐ต are all parallel.

So we know for example that
vector ๐ข runs from ๐ท to ๐ถ, or we can put in vector ๐ข in some various
different places in there as well. So those distances are
parallel, but theyโre also the same length. So they- we can pick the vector
๐ข up and place them in each of those locations. And likewise, vector ๐ฃ,
running from ๐บ to ๐ถ, that will also be vector ๐ฃ; that will also be vector ๐ฃ;
and that will also be vector ๐ฃ.

So we got a few gaps on our
hexagon here. How would I get for example
from ๐บ to ๐ท along this vector here? Well I could go the
straight-line route, but that doesnโt tell me anything in terms of ๐ข and
๐ฃ. So I could also go by this
other route; I could go along from ๐บ to ๐ถ, which is the vector ๐ฃ, and I could
go from ๐ถ to ๐ท, which is the opposite way to ๐ข, so itโs a negative ๐ข.

So vector ๐บ๐ท as we said is
๐บ๐ถ plus ๐ถ๐ท, which is ๐ฃ plus the negative of ๐ข. In other words, ๐ฃ take away
๐ข. So letโs draw that in on the
diagram then: ๐บ๐ท is ๐ฃ take away ๐ข. And likewise, ๐น๐ธ is parallel
and the same length, so that is also ๐ฃ minus ๐ข; A๐บ is too; and so is
๐ต๐ถ.

So when youโre trying to
summarise the journey from ๐ด to ๐ธ in terms of ๐ข and ๐ฃ, so all of these
journeys between individual points on our hexagon are already in terms of ๐ข and
๐ฃ, so we just need to pick a convenient route. So letโs go along here, which
is a negative ๐ข โ itโs the opposite direction to a ๐ข โ and then down here,
which is ๐ฃ minus ๐ข. We better tidy those up. So when we write that out,
weโve got ๐ข plus ๐ฃ minus ๐ข. So when we write that out,
weโve got a negative ๐ข plus ๐ฃ minus ๐ข. And it doesnโt matter what
route we took; however convoluted, wouldโve still come up with that same answer
for ๐ด๐ธ.