Question Video: Expressing Vectors in terms of ๐‘ข and ๐‘ฃ Mathematics

๐ด๐ต๐ถ๐ท๐ธ๐น is a regular hexagon. Express ๐€๐„ in terms of ๐ฎ and ๐ฏ.


Video Transcript

Weโ€™ve got a regular hexagon ๐ด๐ต๐ถ๐ท๐ธ๐น and ๐บ is the midpoint of that and we have to express ๐ด๐ธ in terms of vectors ๐‘ข and ๐‘ฃ. So the vector ๐‘ฃ is from ๐บ to ๐ถ and the vector ๐‘ข is from ๐ท to ๐ถ. Now because this is a regular hexagon, we know that a number of these things are parallel. So ๐ด๐ต and ๐ธ๐ท and ๐น๐บ and ๐บ๐ถ are all parallel; ๐ด๐น and ๐ต๐บ and ๐บ๐ธ and ๐ถ๐ท are all parallel; and ๐ธ๐น, ๐ท๐บ, ๐บ๐ด, and ๐ถ๐ต are all parallel.

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

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

So vector ๐บ๐ท as we said is ๐บ๐ถ plus ๐ถ๐ท, which is ๐‘ฃ plus the negative of ๐‘ข. In other words, ๐‘ฃ take away ๐‘ข. So letโ€™s draw that in on the diagram then: ๐บ๐ท is ๐‘ฃ take away ๐‘ข. And likewise, ๐น๐ธ is parallel and the same length, so that is also ๐‘ฃ minus ๐‘ข; A๐บ is too; and so is ๐ต๐ถ.

So when youโ€™re trying to summarise the journey from ๐ด to ๐ธ in terms of ๐‘ข and ๐‘ฃ, so all of these journeys between individual points on our hexagon are already in terms of ๐‘ข and ๐‘ฃ, so we just need to pick a convenient route. So letโ€™s go along here, which is a negative ๐‘ข โ€” itโ€™s the opposite direction to a ๐‘ข โ€” and then down here, which is ๐‘ฃ minus ๐‘ข. We better tidy those up. So when we write that out, weโ€™ve got ๐‘ข plus ๐‘ฃ minus ๐‘ข. So when we write that out, weโ€™ve got a negative ๐‘ข plus ๐‘ฃ minus ๐‘ข. And it doesnโ€™t matter what route we took; however convoluted, wouldโ€™ve still come up with that same answer for ๐ด๐ธ.

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