O is the origin, OP = p and OQ = q. Z is a point on PQ such that PZ : ZQ = 5 : 2. How to work out, in terms of p and q, the position vector of Z?

Question

I don't understand why the fraction part of vector PZ= 5/7 (-p+q) is 5/7, shouldn't it be 2/7 as the route taken is PO + OQ + QZ?

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Answer 1 · 2018-02-16T09:55:18

see below

#PZ:ZQ::5:7#

so #PZ" "# is #5" "#parts and#" "ZQ" "#is #2" "#parts.

so #PQ" "#is #" "5+2=7" parts altogether"#

#=>PZ=5/7PQ#

we want the position vector of #Z" "# which is #vec(OZ)#

#vec(OZ)=vec(OP)+vec(PZ)#

#vec(OZ)=vec(OP)+5/7vec(PQ)#

#vec(OZ)=vec(OP)+5/7(vec(PO)+vec(OQ))#

#vec(OZ)=vecp+5/7(-vecp+vecq)#

#vec(OZ)+2/7vecp+5/7vecq=1/7(2vecp+5vecq)#

on simplification

the route you have put down

#vec(PO)+vec(OQ)+vec(QZ)#

gives #vec(PZ)#

and not the position vector of #Z " "# which is #vec(OZ)#