Problem 85 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

If \(C\) is the mid point of \(A B\) and \(P\) is any point outside \(A B\), then [2005] (A) \(\overline{P A}+\overline{P B}=2 \overrightarrow{P C}\) (B) \(\overline{P A}+\overrightarrow{P B}=\overrightarrow{P C}\) (C) \(\overrightarrow{P A}+\overrightarrow{P B}+2 \overrightarrow{P C}=0\) (D) \(\overrightarrow{P A}+\overrightarrow{P B}+\overrightarrow{P C}=0\)

Step-by-Step Solution

Answer

Option (C) \(\overrightarrow{P A} + \overrightarrow{P B} + 2\overrightarrow{P C} = 0\).

Step 1: Identify Given Information

We know that \(C\) is the midpoint of \(AB\), which implies that \(C\) divides \(AB\) into two equal segments. Therefore, the position vector \(\vec{C}\) is given by \(\vec{C} = \frac{\vec{A} + \vec{B}}{2}\). We need to find the correct resultant vector relation that holds when \(P\) is any point outside \(AB\).

Step 2: Express Vector PC

Since \(C\) is the midpoint of \(AB\), we can represent \(\vec{C}\) as \(\frac{\vec{A} + \vec{B}}{2}\). Thus, vector \(\vec{PC}\) can be written as \(\vec{PC} = \vec{C} - \vec{P}\). Substitute \(\vec{C}\) to get \(\vec{PC} = \frac{\vec{A} + \vec{B}}{2} - \vec{P}\).

Step 3: Calculate Vector PA and PB

Define vectors \(\vec{PA}\) as \(\vec{A} - \vec{P}\) and \(\vec{PB}\) as \(\vec{B} - \vec{P}\).

Step 4: Formulate Combination of Vectors PA, PB, and PC

We are interested in the combination of \(\vec{PA} + \vec{PB} + 2\vec{PC}\). Using the expressions from earlier steps: \(\vec{PA} = \vec{A} - \vec{P}\), \(\vec{PB} = \vec{B} - \vec{P}\), and \(\vec{PC} = \frac{\vec{A} + \vec{B}}{2} - \vec{P}\).

Step 5: Solve and Simplify the Expression

Plug the vectors into the combination: \(\vec{PA} + \vec{PB} + 2\vec{PC} = (\vec{A} - \vec{P}) + (\vec{B} - \vec{P}) + 2\left( \frac{\vec{A} + \vec{B}}{2} - \vec{P} \right)\). Simplifying, this becomes \(\vec{A} + \vec{B} - 2\vec{P} + \vec{A} + \vec{B} - 2\vec{P}\). Combine like terms, and you get \(2\vec{A} + 2\vec{B} - 4\vec{P} + \vec{A} + \vec{B} - 2\vec{P} = 0\).

Step 6: Final Expression

The expression simplifies to \(\vec{PA} + \vec{PB} + 2\vec{PC} = 0\). This corresponds to option \( (C) \) \(\overrightarrow{P A} + \overrightarrow{P B} + 2\overrightarrow{P C} = 0\).

Key Concepts

Midpoint Theorem

The Midpoint Theorem is a fundamental concept in geometry that revolves around dividing a line segment into two equal parts. When a point, called the midpoint, lies exactly halfway between the endpoints of a line segment, it divides the segment into two equal lengths. This principle is not only useful in geometric constructions but also plays a significant role in vector analysis.

In vector algebra, the midpoint theorem can be expressed using position vectors. If the endpoints of the segment are represented by position vectors \( \vec{A} \) and \( \vec{B} \), the position vector of the midpoint \( C \) is found using the formula:

• \( \vec{C} = \frac{\vec{A} + \vec{B}}{2} \)

This equation signifies that vector \( \vec{C} \) is the average of vectors \( \vec{A} \) and \( \vec{B} \). Thus, it establishes \( C \) as the midpoint by ensuring that \( C \) is equidistant from \( A \) and \( B \).

Understanding the midpoint theorem is crucial for solving problems that involve bisecting segments or finding equilibrium points in vector fields.

Position Vectors

Position vectors are a straightforward yet powerful tool in mathematics, particularly in vector algebra. These vectors originate from a common reference point, often chosen as the origin of the coordinate system, and point towards a specific location in space, representing the positional information of that point.

The position vector of a point \( A \) is usually denoted by \( \vec{A} \) and can be visualized as an arrow originating from the origin pointing towards the location of point \( A \). It is defined by its coordinates \((x, y, z)\) in a Cartesian coordinate system.

• Position vectors provide a method to describe the location and orientation of points in various dimensions.
• They are crucial for performing vector operations such as addition, subtraction, and scalar multiplication.

In the given problem, the position vectors of the points \( A \), \( B \), and \( C \) are used to determine relationships between vectors associated with the midpoint and other points. Grasping the concept of position vectors helps in visualizing geometric problems and aids in the simplification of complex vector expressions.

Vector Addition

Vector addition is a key operation in vector algebra that combines two or more vectors to produce another vector. This operation is essential for determining resultant vectors, which signify the combined effect of multiple vector quantities.

To perform vector addition, the corresponding components of the vectors are summed. If you have two vectors \( \vec{A} = (a_1, a_2, a_3) \) and \( \vec{B} = (b_1, b_2, b_3) \), their sum \( \vec{A} + \vec{B} \) is:

• \( \vec{A} + \vec{B} = (a_1 + b_1, a_2 + b_2, a_3 + b_3) \)

This operation follows the parallelogram law, where the sum of two vectors geometrically forms the diagonal of a parallelogram constructed with the two vectors as adjacent sides.

In the context of the exercise, vector addition is utilized to combine the effects of \( \vec{PA} \), \( \vec{PB} \), and \( 2\vec{PC} \). This results in determining their collective impact, helping to solve for conditions like equilibrium or midpoint-related properties. Mastery of vector addition allows for precise navigation and formulation of vector-related problems.