Question Video: Expressing Vectors in terms of 𝑢 and 𝑣 | Nagwa Question Video: Expressing Vectors in terms of 𝑢 and 𝑣 | Nagwa

# Question Video: Expressing Vectors in terms of ๐ข and ๐ฃ Mathematics • First Year of Secondary School

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

02:38

### 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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