Question

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?

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?

Answer 1

see below

Explanation:

`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)`