If \(3 \hat{\mathrm{j}}, 4 \hat{\mathrm{k}}\) and \(3 \hat{\mathrm{j}}+4 \hat{\mathrm{k}}\) are the position vectors of the vertices \(\mathrm{A}, \mathrm{B}\), \(C\) respectively of \(\triangle \mathrm{ABC}\), then the position vector of the point in which the bisector of \(\angle \mathrm{A}\) meets \(\mathrm{BC}\) is

We have \(\overline{\mathrm{a}}=3 \hat{\mathrm{j}}, \mathrm{b}=4 \hat{\mathrm{k}}\) and \(\overline{\mathrm{c}}=3 \hat{\mathrm{j}}+4 \hat{\mathrm{k}}\)
Angle bisector \(\mathrm{AD}\) divides \(\mathrm{BC}\) in the ratio \(\mathrm{AB}: \mathrm{AC}\)
\(
\mathrm{AB}=\sqrt{9+16}=5 \text { and } \mathrm{AC}=\sqrt{0+16}=4
\)
Thus \(\mathrm{D}\) divides \(\mathrm{BC}\) in the ratio \(5: 4\)
\(
\begin{aligned}
& \therefore \mathrm{D}=\left[0, \frac{5(3)}{5+4}, \frac{5(4)+4(4)}{5+4}\right] \\
& =\left(0, \frac{15}{9}, \frac{20+16}{9}\right)=\left(0, \frac{5}{3}, 4\right)=\frac{5}{3} \hat{\mathrm{j}}+4 \hat{\mathrm{k}}
\end{aligned}
\)