If \(A(0,4,0), \quad B(0,0,3)\) and \(C(0,4,3)\) are the vertices of \(\Delta A B C\), then its incentre is

Let
\(\overline{\mathrm{a}}=4 \hat{\mathrm{j}} \therefore|\overline{\mathrm{a}}|=\mathrm{a}=4 \)
\(\overline{\mathrm{b}}=3 \hat{\mathrm{k}} \therefore|\overline{\mathrm{b}}|=\mathrm{b}=3 \)
\(\overline{\mathrm{c}}=4 \hat{\mathrm{j}}+3 \hat{\mathrm{k}} \therefore|\overline{\mathrm{c}}|=\mathrm{c}=\sqrt{16+9}=5\)
Let \(H(\bar{h})\) be the incentre
Incentre is given by \(\frac{a \bar{a}+b \bar{b}+c \bar{c}}{a+b+c}=\bar{h} \)
\(\Rightarrow \bar{h}=\frac{4(4 \hat{j})+3(3 \hat{k})+5(4 \hat{j}+3 \hat{k})}{4+3+5} \)
\(=\frac{16 \hat{j}+9 \hat{k}+20 \hat{j}+15 \hat{k}}{12}=\frac{36 \hat{j}+24 \hat{k}}{12}=3 \hat{j}+2 \hat{k}\)
Thus co-ordinate of incentre is \((0,3,2)\)