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

The position vectors `bar(p),bar(q),bar(r)` of the vertices P,Q,R are
`bar(p)=4hat(j),bar(q)=3hat(k)andbar(r)=4hat(j)+3hat(k)`
`:.bar(PQ)=bar(q)=3hat(k)-4hat(j)`
`=-4hat(j)+3hat(k)`
`bar(QR)=bar(r)-bar(q)=(4hat(j)+3hat(k))-(3hat(k))=4hat(j)`
`andbar(PR)=bar(r)-bar(p)=(4hat(j)+3hat(k))=4hat(j)=3hat(k)`
Let `x=|bar(QR)|,y=|bar(PR)|andz=|bar(PQ)|`
`:.x=sqrt(0+4^(2)+0)=4`
`y=sqrt(0+0+3^(2))=3`
`andz=sqrt(0+(-4)^(2)+3^(2))=5`
If `H(bar(h))` is the incentre of the triangle PQR, then
`bar(h)=(xbar(p)+ybar(q)+zbar(r))/(x+y+z)`
`=(4(4bar(j))+3(3bar(k))+5(4bar(j)+3bar(k)))/(4+3+5)`
`=(1)/(12)(16hat(j)+9hat(k)+20hat(j)+15hat(k))=(1)/(12)(36hat(j)+24hat(k))`
`:.bar(h)=3hat(j)+2hat(k)" ":.H=(0,3,2)`.