### Video Transcript

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.