If the position vectors of the vertices \(A, B, C\) of a tringle \(A B C\) are \(4 \hat{\imath}+7 j+8 \hat{k}, 2 \hat{\imath}+3 \hat{\jmath}+4 \hat{k}\) and \(2 \hat{\imath}+5 \hat{\jmath}+7 \hat{k}\) respectively, then the position vector of the point where bisector of angle A meets \(B C\) is

Suppose the bisector of angle \(\mathrm{A}\) meets \(\mathrm{BC}\) at \(\mathrm{D}\).
Then \(A D\) divides \(B C\) in the ratio \(A B\) : \(A C\)
\(\therefore\) Position vector of \(\mathrm{D}=\frac{(\mathrm{AC})(2 \hat{\mathrm{i}}+3 \hat{\mathrm{j}}+4 \hat{\mathrm{k}})+(\mathrm{AB})(2 \hat{\mathrm{i}}+5 \hat{\mathrm{j}}+7 \hat{\mathrm{k}})}{(\mathrm{AB})+(\mathrm{AC})}\)
\(\overline{\mathrm{AB}}=-2 \hat{\mathrm{i}}-4 \hat{\mathrm{j}}-4 \hat{\mathrm{k}}\) and \(\overline{\mathrm{AC}}=-2 \hat{\mathrm{i}}-2 \hat{\mathrm{j}}-\hat{\mathrm{k}}\)
\(|\overline{\mathrm{AB}}|=\mathrm{AB}=6\) and \(|\overline{\mathrm{AC}}|=\mathrm{AC}=\sqrt{4+4+1}=3\)
Position vector of \(D\)
\(=\frac{6(2 \hat{i}+3 \hat{j}+4 \hat{k})+3(2 \hat{i}+5 \hat{j}+7 \hat{k})}{6+3}=\frac{18 \hat{i}+33 \hat{j}+45 \hat{k}}{9}=\) \(\frac{1}{3}(6 \hat{i}+11 \hat{j}+15 \hat{k})\)