Using vector method, find the incentre of the triangle whose vertices are P(0,4,0),Q(0,0,3) and R(0,4,3).

Explanation

Let `barp , bar q, barr`, be the position vectro of the points P,Q,R respectively.
`:. barp= 4hatj,`
`barq=3hatk,`
`barr=4hatj+3hatk`
Consider `bar (PQ)=barq-barp=3hatk-4hatj`
`=-4hatj+3hatk`
`|bar(PQ)|=sqrt((-4)^(2)+3^(2))=sqrt(16+9)`
`sqrt(25)=5`
`|bar(PQ)|=sqrt((-4)^(2)+3^(2))=sqrt(16+9)`
`bar(QR)= bar r-barq=4hatj+3hatk-3hatk`
`=4hatj`
`|bar(QR)|=sqrt(4^(2))=4`
`bar(PR)=barr-barp=4hatj+3hatj-4hatj`
`=3hatk`
`:. |bar(PR)|=sqrt(3^(2))=3`
Let `|bar(QR)|=x=4`
`|bar(PR)|=y=3 and |bar(PQ)|`
`=z=5`
If `h(barh)` is the incentre of `Delta PQR` , then
`bar h=(xbarp+ybarq+zbarr)/(x+y+z)`
`=(4*(4hatj)+3*(3hatk)+5*(4hatj+3hatk))/(4+3+5)`
`=(16hatj+9hatk+20hatj+15hatk)/(12)`
`(36hatj+24hatk)/(12)=3hatj+2hatk`
`:. H-=(0,3,2)` is the incentre of `Delta PQR`