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

Question

If \(C\) is the mid point of \(A B\) and \(P\) is any point outside \(A B\), then (A) \(\overrightarrow{P A}+\overrightarrow{P B}=2 \overrightarrow{P C}\) (B) \(\overrightarrow{P A}+\overrightarrow{P B}=\overrightarrow{P C}\) (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 (A) is correct: \(\overrightarrow{P A} + \overrightarrow{P B} = 2 \overrightarrow{P C}\).

Step 1: Understand the Midpoint

Given a line segment \(AB\) with \(C\) as the midpoint, the position vector of \(C\) can be represented as the average of position vectors of \(A\) and \(B\). Thus, \(\overrightarrow{C} = \frac{1}{2}(\overrightarrow{A} + \overrightarrow{B})\).

Step 2: Relate Vectors with Point P

Since \(P\) is a point outside \(AB\), analyze the vector relations given in the options. To simplify, consider expressions involving \(\overrightarrow{P C}\) in terms of \(\overrightarrow{P A}\) and \(\overrightarrow{P B}\) using \(C\) as the midpoint.

Step 3: Calculate \(\overrightarrow{P C}\)

\(\overrightarrow{P C} = \overrightarrow{P A} + \overrightarrow{A C}\). Due to \(C\) being the midpoint, \(\overrightarrow{A C} = \frac{1}{2} \overrightarrow{A B}\). So, \(\overrightarrow{P C} = \overrightarrow{P A} + \frac{1}{2} \overrightarrow{A B}\).

Step 4: Expression for \(\overrightarrow{P A} + \overrightarrow{P B}\)

From the definitions, we know \(\overrightarrow{P B} = \overrightarrow{P A} + \overrightarrow{A B}\). Therefore, \(\overrightarrow{P A} + \overrightarrow{P B} = \overrightarrow{P A} + (\overrightarrow{P A} + \overrightarrow{A B}) = 2\overrightarrow{P A} + \overrightarrow{A B}\).

Step 5: Simplification Using \(\overrightarrow{P C}\)

Substitute \(\overrightarrow{P C} = \overrightarrow{P A} + \frac{1}{2} \overrightarrow{A B}\) into the equation to find: \(2\overrightarrow{P C} = 2(\overrightarrow{P A} + \frac{1}{2}\overrightarrow{A B}) = 2\overrightarrow{P A} + \overrightarrow{A B}\). Thus, \(\overrightarrow{P A} + \overrightarrow{P B} = 2\overrightarrow{P C}\).

Step 6: Conclusion

The vector equation derived matches option (A). Therefore, \(\overrightarrow{P A} + \overrightarrow{P B} = 2 \overrightarrow{P C}\).

Key Concepts

Midpoint Theorem

The midpoint theorem is a fundamental concept in vector algebra, primarily used in geometry. It states that if a point is the midpoint of a line segment, then the line segment is divided into two equal parts. For any line segment \(AB\) with midpoint \(C\), the position vector of \(C\) can be expressed as the average of the position vectors of \(A\) and \(B\). Mathematically, this relationship is shown as:

\[ \overrightarrow{C} = \frac{1}{2}(\overrightarrow{A} + \overrightarrow{B}) \]
This equation illustrates how \(C\) is equidistant from both \(A\) and \(B\). Understanding this principle helps to simplify vector calculations, especially when dealing with vector sums and differences. It is particularly useful when analyzing vector relationships in geometric contexts, like the line segment problem, where \(C\) is used to determine properties of other vectors, such as \(\overrightarrow{PC}\).

Vector Algebra

Vector algebra involves operations like vector addition, subtraction, and scalar multiplication. These operations follow specific rules that make vector calculations uniform and predictable.

For vector addition, we add vectors by putting them head-to-tail, forming a resultant vector from the tail of the first vector to the head of the last. This is essential when combining components like \(\overrightarrow{PA}\) and \(\overrightarrow{PB}\) in our exercise, where they combine to form a new vector. In the expression \(\overrightarrow{PA} + \overrightarrow{PB} = 2\overrightarrow{PC}\), we see an application of vector addition.

Vector algebra also extends to linear combinations of vectors, allowing us to express one vector in terms of others using scalar multiplication. In this exercise, knowing \(\overrightarrow{PC} = \overrightarrow{PA} + \frac{1}{2}\overrightarrow{AB}\) demonstrates a component form derived from multiplying \(\overrightarrow{AB}\) by a scalar \(\frac{1}{2}\), directly linking it to the midpoint concept.

Geometric Vectors

Geometric vectors are visual representations of vectors in geometry. They are defined by both direction and magnitude and are often drawn as arrows. In geometric contexts, vectors represent quantities that have both size and orientation, like displacement or force.

In the given exercise, each vector, such as \(\overrightarrow{PA}\) or \(\overrightarrow{PB}\), can be visualized as arrows originating from a point \(P\), pointing toward points \(A\) or \(B\). Recognizing these as geometric vectors helps understand their physical implications and aids in the graphical depiction of vector operations.

When considering the sum of vectors like \(\overrightarrow{PA} + \overrightarrow{PB}\), the geometric interpretation reinforces our understanding of the midpoint theorem and vector algebra. The sums thus transform abstract mathematical concepts into tangible elements, easily interpreted through geometric vectors.