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#.

1 Answer
Douglas K.
Mar 18, 2018

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#

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