Problem 9 - A Complete Resource Book in Mathematics for JEE Main

Question

\(A, B, C, D\) are four points on a plane with position vectors \(a, b, c, d\), respectively, such that \((a-d)\). \((b-c)=(b-d) \cdot(c-a)=0 .\) For \(\Delta A B C, D\) is the (A) incentre (B) orthocentre (C) centroid (D) none of these

Step-by-Step Solution

Answer

(B) orthocentre

Step 1: Understand Given Conditions

We are given four points on a plane with their respective position vectors \(a, b, c, d\). There are two conditions: \((a-d) \cdot (b-c) = 0\) and \((b-d) \cdot (c-a) = 0\). These conditions need to be satisfied for any specific geometric center.

Step 2: Analyze the First Condition

The first condition \((a-d) \cdot (b-c) = 0\) states that the vector \(a-d\) is perpendicular to the vector \(b-c\). This indicates that line AD is perpendicular to line BC.

Step 3: Analyze the Second Condition

The second condition \((b-d) \cdot (c-a) = 0\) states that the vector \(b-d\) is perpendicular to the vector \(c-a\). This means line BD is perpendicular to line CA.

Step 4: Deduce the Role of D in Triangle ABC

Since AD is perpendicular to BC and BD is perpendicular to CA, point D lies on the intersection of the altitudes of \(\Delta ABC\). This intersection is called the orthocenter of the triangle.

Step 5: Confirm D as the Orthocenter

The orthocenter is where the altitudes of the triangle meet. From the given conditions, D satisfies this definition as it is determined by the perpendicularity of the vectors connecting points A, B, and C.

Step 6: Verify the Correct Option

Based on our deductions, D is the orthocenter of triangle ABC. Thus, the answer to the problem is consistent with option (B).

Key Concepts

Perpendicularity

Perpendicularity is a key concept in geometry and vector analysis. It refers to the relationship between two lines or vectors that intersect at a right angle, or 90 degrees.
In the problem, we saw two conditions involving dot products of vectors which help define perpendicularity:

• The condition \( (a-d) \cdot (b-c) = 0 \) implies that vector \(a-d\) is perpendicular to vector \(b-c\). This reflects in the perpendicularity of line AD with line BC.
• Similarly, \( (b-d) \cdot (c-a) = 0 \) suggests that vector \(b-d\) is perpendicular to \(c-a\), indicating line BD is perpendicular to line CA.

In terms of geometry, these perpendicular vectors are reminiscent of the concept of heights or altitudes falling perpendicularly from vertices to the opposite sides.
Understanding these perpendicular relationships is vital in determining specific points like orthocenters in triangles.

Altitudes

Altitudes in a triangle are critical lines that are perpendicular to one side, or base, and pass through the opposite vertex, often intersecting at a notable point called the orthocenter.
In our geometric problem:

• The line AD acts as an altitude since it is perpendicular to line BC.
• Similarly, line BD serves as another altitude as it intersects line CA perpendicularly.

What makes altitudes fascinating is that all three existent altitudes in any triangle meet at a single point known as the orthocenter. This contrasts with the term 'perpendicular bisector', which would only bisect the side. Altitudes are concerned with the perpendicular intersection from one vertex to its opposite side.
By tracking the paths of these altitudes, we confirm where these special intersection points like the orthocenter lie.

Position Vectors

Position vectors are mathematical tools used to denote the location of points in a coordinate plane. They are expressed as directed line segments from the origin to a specific point.
In the exercise, points A, B, C, and D are represented with position vectors \(a, b, c, d\). These vectors help us calculate connections between pairs of points:

• By analyzing vectors like \(a-d\) and \(b-c\), we deduce geometrical properties such as the perpendicularity conditions.
• Having vectors allows for mathematical operations, such as the dot product, to be used in confirming relationships like orthogonality.

The interpretation of vectors not only conveys direction and magnitude, it also simplifies understanding spatial relationships in more efficient numerical formats. Comprehending position vectors enables us to effectively detail and articulate points in geometric arrangements like triangulation and altitudinal determination.