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

Engineering

Mathematics

Vector Addition and Subtraction

Section Formulae and Centres of a Triangle

Circumradius Inradius and exradius

Question

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)

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\vec{P} = 4 \hat{j}

\vec{Q} = 3 \hat{k}

\vec{R} = 4 \hat{j} + 3 \hat{k}

\vec{P Q} = \vec{Q} - \vec{P} = - 4 \hat{j} + 3 \hat{k}

\vec{Q R} = \vec{R} - \vec{Q} = 4 \hat{j}

\vec{R} P = \vec{P} - \vec{R} = - 3 \hat{k}

∣ \vec{P Q} ∣ = 5, ∣ \vec{Q R} ∣ = 4, ∣ \vec{R P} ∣ = 3

The incenter is the intersection of three angle bisectors,

∴ \vec{I} = \frac{∣ \vec{Q R} ∣ \cdot \vec{P} + ∣ \vec{R P} ∣ \cdot \vec{Q} + ∣ \vec{P Q} ∣ \cdot \vec{R}}{∣ \vec{P Q} ∣ + ∣ \vec{Q R} ∣ + ∣ \vec{R P} ∣}

\Rightarrow \vec{I} = \frac{16 \hat{j} + 9 \hat{k} + 20 \hat{j} + 15 \hat{k}}{12}

\Rightarrow \vec{I} = 3 \hat{j} + 2 \hat{k}

∴,

Coordinates of incentre are

(0, 3, 2)