IIT-JEE 1984Question 1
$A,B,C$ and $D$ are four points in a plane with position vectors $\vec a,\vec b,\vec c$ and $\vec d$ respectively such that $(\vec a-\vec d)\cdot(\vec b-\vec c)=(\vec b-\vec d)\cdot(\vec c-\vec a)=0$. The point $D$, then, is the ______ of the triangle $ABC$.
Vector Algebra
Answerorthocentre
Solution
$(\vec a-\vec d)\cdot(\vec b-\vec c)=0$ gives $DA\perp BC$, and $(\vec b-\vec d)\cdot(\vec c-\vec a)=0$ gives $DB\perp CA$. $D$ lies on two altitudes, so it is the orthocentre.