Vector Algebra

Consider triangle ABC with position vectors .
Altitude from A to BC: Perpendicular to .
Let foot be D. : .
Orthocenter H satisfies: , , .
Solve: Let , , .
The equations are linear in , and the system has a unique solution, proving concurrence.
Answer: Proved using perpendicularity conditions.

The vector equation of a plane through point with normal perpendicular to vectors and is .

Point : .

Vectors: , .

Normal vector: .

Plane equation: .

Or, normal form: .

Vector equation: .

Answer: .

Let divide internally in ratio . By section formula:
Position vector of :
Proof using vectors:
Since divides in , .
Answer: Proved.