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…

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
  1. $\frac{1}{3}(6 \hat{\imath}+11 \hat{\jmath}+15 \hat{\mathrm{k}})$
  2. $\frac{1}{2}(4 \hat{\imath}+8 \hat{\jmath}+11 \hat{\mathrm{k}})$
  3. $\frac{1}{4}(8 \hat{\imath}+14 \hat{\jmath}+19 \hat{\mathrm{k}})$
  4. $\frac{1}{3}(6 \hat{\imath}+13 \hat{\mathbf{j}}+18 \hat{\mathrm{k}})$

Solution

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 $D$ $=\frac{6(2 \hat{i}+3 \hat{j}+4 \hat{k})+3(2 \hat{i}+5 \hat{j}+7 \hat{k})}{6+3}=\frac{18 \hat{i}+33 \hat{j}+45 \hat{k}}{9}=\frac{1}{3}(6 \hat{i}+11 \hat{j}+15 \hat{k})$

Asked in: MHT CET 2020 (20 Oct Shift 1)

Practice more Vectors questions on Aicharya