Video Transcript
We’ve got a regular hexagon
𝐴𝐵𝐶𝐷𝐸𝐹 and 𝐺 is the midpoint of that and we have to express 𝐴𝐸 in terms
of vectors 𝑢 and 𝑣. So the vector 𝑣 is from 𝐺 to
𝐶 and the vector 𝑢 is from 𝐷 to 𝐶. Now because this is a regular
hexagon, we know that a number of these things are parallel. So 𝐴𝐵 and 𝐸𝐷 and 𝐹𝐺 and
𝐺𝐶 are all parallel; 𝐴𝐹 and 𝐵𝐺 and 𝐺𝐸 and 𝐶𝐷 are all parallel; and
𝐸𝐹, 𝐷𝐺, 𝐺𝐴, and 𝐶𝐵 are all parallel.
So we know for example that
vector 𝑢 runs from 𝐷 to 𝐶, or we can put in vector 𝑢 in some various
different places in there as well. So those distances are
parallel, but they’re also the same length. So they- we can pick the vector
𝑢 up and place them in each of those locations. And likewise, vector 𝑣,
running from 𝐺 to 𝐶, that will also be vector 𝑣; that will also be vector 𝑣;
and that will also be vector 𝑣.
So we got a few gaps on our
hexagon here. How would I get for example
from 𝐺 to 𝐷 along this vector here? Well I could go the
straight-line route, but that doesn’t tell me anything in terms of 𝑢 and
𝑣. So I could also go by this
other route; I could go along from 𝐺 to 𝐶, which is the vector 𝑣, and I could
go from 𝐶 to 𝐷, which is the opposite way to 𝑢, so it’s a negative 𝑢.
So vector 𝐺𝐷 as we said is
𝐺𝐶 plus 𝐶𝐷, which is 𝑣 plus the negative of 𝑢. In other words, 𝑣 take away
𝑢. So let’s draw that in on the
diagram then: 𝐺𝐷 is 𝑣 take away 𝑢. And likewise, 𝐹𝐸 is parallel
and the same length, so that is also 𝑣 minus 𝑢; A𝐺 is too; and so is
𝐵𝐶.
So when you’re trying to
summarise the journey from 𝐴 to 𝐸 in terms of 𝑢 and 𝑣, so all of these
journeys between individual points on our hexagon are already in terms of 𝑢 and
𝑣, so we just need to pick a convenient route. So let’s go along here, which
is a negative 𝑢 — it’s the opposite direction to a 𝑢 — and then down here,
which is 𝑣 minus 𝑢. We better tidy those up. So when we write that out,
we’ve got 𝑢 plus 𝑣 minus 𝑢. So when we write that out,
we’ve got a negative 𝑢 plus 𝑣 minus 𝑢. And it doesn’t matter what
route we took; however convoluted, would’ve still come up with that same answer
for 𝐴𝐸.
