Let the position vectors of the vertices A,B and C of a triangle be 2i+2j+k,i+2j+2k and 2i…

Question

Let the position vectors of the vertices $A,B$ and $C$ of a triangle be $2\hat{i}+2\hat{j}+\hat{k},\hat{i}+2\hat{j}+2\hat{k}$ and $2\hat{i}+\hat{j}+2\hat{k}$ respectively. Let ${l}_{1},{l}_{2}$ and ${l}_{3}$ be the lengths of perpendiculars drawn from the ortho center of the triangle on the sides $AB,BC$ and $CA$ respectively, then ${l}_{2}^{1}+{l}_{2}^{2}+{l}_{2}^{3}$ equals:

Accepted Answer

$	riangleABC$ is equilateral

Orthocentre and centroid will be same

$G(\frac{5}{3},\frac{5}{3},\frac{5}{3})$

Mid-point of $AB$ is $D(\frac{3}{2},2,\frac{3}{2})$

$\begin{gathered} 	herefore {l}_{1}=\sqrt{\frac{1}{36}+\frac{1}{9}+\frac{1}{36}} \ {l}_{1}=\sqrt{\frac{1}{6}}={l}_{2}={l}_{3} \ 	herefore {l}_{2}^{1}+{l}_{2}^{2}+{l}_{2}^{3}=\frac{1}{2} \end{gathered}$

Solutions · 1