Let the position vectors of the vertices $A, B$ and $C$ of a tetrahedron $A B C D$ be $\hat{\mathbf{i}}+2…

Let the position vectors of the vertices $A, B$ and $C$ of a tetrahedron $A B C D$ be $\hat{\mathbf{i}}+2 \hat{\mathbf{j}}+\hat{\mathbf{k}}, \hat{\mathbf{i}}+3 \hat{\mathbf{j}}-2 \hat{k}$ and $2 \hat{i}+\hat{j}-\hat{k}$ respectively. The altitude from the vertex $D$ to the opposite face $A B C$ meets the median line segment through $A$ of the triangle $A B C$ at the point $E$. If the length of $A D$ is $\frac{\sqrt{110}}{3}$ and the volume of the tetrahedron is $\frac{\sqrt{805}}{6 \sqrt{2}}$, then the position vector of $E$ is
  1. $\frac{1}{12}(7 \hat{\mathbf{i}}+4 \hat{\mathbf{j}}+3 \hat{k})$
  2. $\frac{1}{2}(\hat{i}+4 \hat{j}+7 \hat{k})$
  3. $\frac{1}{6}(12 \hat{i}+12 \hat{j}+\hat{k})$
  4. $\frac{1}{6}(7 \hat{\mathrm{i}}+12 \hat{\mathrm{j}}+\hat{\mathrm{k}})$

Solution


Area of $\triangle \mathrm{ABC}=\frac{1}{2}|\overrightarrow{\mathrm{AB}} \times \overrightarrow{\mathrm{AC}}|$
$=\frac{1}{2}|5 \hat{\mathrm{i}}+3 \hat{\mathrm{j}}+\hat{\mathrm{k}}|=\frac{1}{2} \sqrt{35}$
volume of tetrahedron
$\begin{aligned}
& =\frac{1}{3} \times \text { Base area } \times \mathrm{h}=\frac{\sqrt{805}}{6 \sqrt{2}} \\ & \frac{1}{3} \times \frac{1}{2} \sqrt{35} \times \mathrm{h}=\frac{\sqrt{805}}{6 \sqrt{2}} \\ & \mathrm{~h}=\sqrt{\frac{23}{2}}
\end{aligned}$
$\mathrm{AE}^2=\mathrm{AD}^2-\mathrm{DE}^2=\frac{13}{18} \therefore \mathrm{AE}=\sqrt{\frac{13}{18}}$
$\begin{aligned} & \overrightarrow{\mathrm{AE}}=|\mathrm{AE}| \cdot\left(\frac{\hat{\mathrm{i}}-5 \hat{\mathrm{k}}}{\sqrt{26}}\right) \\ & =\sqrt{\frac{13}{18}} \cdot\left(\frac{\hat{\mathrm{i}}-5 \hat{\mathrm{k}}}{\sqrt{26}}\right) \\ & =\sqrt{\frac{13}{18}} \cdot\left(\frac{\hat{\mathrm{i}}-5 \hat{\mathrm{k}}}{\sqrt{26}}\right)=\frac{\hat{\mathrm{i}}-5 \hat{\mathrm{k}}}{6} \\ & \text { P.V. of } \mathrm{E}=\frac{\hat{\mathrm{i}}-5 \hat{\mathrm{k}}}{6}+\hat{\mathrm{i}}+2 \hat{\mathrm{j}}+\hat{\mathrm{k}}=\frac{1}{6}(7 \hat{\mathrm{i}}+12 \hat{\mathrm{j}}+\hat{\mathrm{k}})\end{aligned}$ ,

Asked in: JEE Main 2025 (23 Jan Shift 1)

Practice more Vectors questions on Aicharya