Question Video: Expressing Vectors in terms of 𝑢 and 𝑣 Mathematics • 12th Grade

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.