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 $vec(OZ)$