Question

How to express these vectors in terms of p and q?

ABCDEF is a regular hexagon with centre O. Suppose `vec(AB) = vecp` and `vec(AF) = vecq`, express `vec(AO)`, `vec(AC)`, `vec(AE)`, `vec(CE)` in terms of `vecp` and `vecq`.

Answer 1

Begin by making a drawing of the 7 points and the two vectors.

Explanation:

The following drawing show points a `color(red)("ABCDEFO")`, `color(blue)(vec(p))`, and `color(green)(vecq)`

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Please observe that vectors `vecp` and `vecq` are tail-to-tail. If you move `vecq` so that its tail is touching the nose of `vecp`, you obtain `vec(AO)`

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This is how you add two vectors, therefore, the equation for `vec(AO)` must be:

`vec(AO) = vecp+vecq`

If we make another copy of `vecp` and place its tail to the nose of `vec(AO)`, we arrive and point C

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This means that:

`vec(AC) = vec(AO) + vecp`

Substitute `vec(AO) = vecp+vecq`:

`vec(AC) = vecp+ vecq + vecp`

`vec(AC) = 2vecp + vecq`

We shall remove the extra copy of `vecp` add an extra copy of `vecq` and place its tail at the nose of the existing copy of `vecq` to arrive at point E:

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This means that:

`vec(AE) = vec(AO) + vecq`

Substitute `vec(AO) = vecp+vecq`:

`vec(AE) = vecp + vecq + vecq`

`vec(AE) = vecp+2vecq`

We do not need a drawing for `vec(CE)`. By the rules of vector addition, the following equation must true:

`vec(AE) = vec(AC)+vec(CE)`

Solve for `vec(CE)`:

`vec(CE) = vec(AE)-vec(AC)`

Substitute `vec(AE) = vecp+2vecq`:

`vec(CE) = vecp+2vecq-vec(AC)`

Substitute `vec(AC) = 2vecp + vecq`:

`vec(CE) = vecp+2vecq-(2vecp + vecq)`

`vec(CE) = vecq-vecp`