Question:
<p>\(A,B,C,D\) are four points in a plane with position vectors \(\vec{a},\vec{b},\vec{c},\vec{d}\) respectively such that \((\vec{a}-\vec{d})\cdot(\vec{b}-\vec{c})=0\) and \((\vec{b}-\vec{d})\cdot(\vec{c}-\vec{a})=0\). Then \(D\) is the</p>
centroid of \(\triangle ABC\)
circumcenter of \(\triangle ABC\)
orthocenter of \(\triangle ABC\)
incentre of \(\triangle ABC\)
Step-by-Step Solution
Key Concept: The conditions (a-d) \cdot (b-c)=0 means DA\perpBC, and (b-d) \cdot (c-a)=0 means DB\perpCA. The point from which perpendiculars to all sides pass is the orthocenter.
Condition 1: $\overrightarrow{DA}\cdot\overrightarrow{BC}=0\Rightarrow DA\perp BC$.
Condition 2: $\overrightarrow{DB}\cdot\overrightarrow{CA}=0\Rightarrow DB\perp CA$.
A point from which the two altitudes DA (to BC) and DB (to CA) pass is the orthocenter of △ABC.
Answer: (D) orthocenter.
Correct Answer: D