# Using vector method, find incentre of the triangle whoose vertices are P(0, 4, 0), Q(0, 0, 3) and R(0, 4, 3). - Mathematics and Statistics

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

#### Solution

Let barp, barq ,barr be the position vectors of vertices P, Q, R of Δ PQR respectively

barp=4hatj,barq=3hatk,barr=4hatj+3hatk

bar(PQ)=barq-barp=3hatk-4hatj=-4hatj+3hatk

bar(QR)=barr-barq=4hatj+3hatk-3hatk=4hatj

bar(RP)=barp-barr=4hatj-4hatj-3hatk=-3hatk

Let x, y, z be the lengths of opposites of vertices P,Q,R respectively.

x=|bar(QR)|=4

y=|bar(RP)|=3

z=|bar(PQ)|=sqrt(16+9)=sqrt25=5

If H(barh)is the incentre of DeltaPQR then

barh=(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

Concept: Vectors - Application of Vectors to Geometry
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