Problem 21 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

If \(\mathrm{C}\) is the mid point of \(\mathrm{AB}\) and \(\mathrm{P}\) is any point outside \(\mathrm{AB}\), then \([\mathbf{2 0 0 5}]\) (a) \(\overrightarrow{P A}+\overrightarrow{P B}=2 \overrightarrow{P C}\) (b) \(\overrightarrow{P A}+\overrightarrow{P B}=\overrightarrow{P C}\) (c) \(\overline{P A}+\overrightarrow{P B}+2 \overrightarrow{P C}=\overrightarrow{0}\) (d) \(\overrightarrow{P A}+\overrightarrow{P B}+\overrightarrow{P C}=\overrightarrow{0}\)

Step-by-Step Solution

Answer

The correct option is (a) \(\overrightarrow{PA} + \overrightarrow{PB} = 2\overrightarrow{PC}\).

Step 1: Identify the given information

We are told that \(C\) is the midpoint of \(AB\) and \(P\) is a point outside \(AB\). From this, we recognize that if \(C\) is the midpoint, \(AC = CB\).

Step 2: Express vector relationships using mid-point

Since \(C\) is the midpoint of \(AB\), we can express \(C\) in terms of \(A\) and \(B\) as \(\overrightarrow{C} = \frac{1}{2} (\overrightarrow{A} + \overrightarrow{B})\).

Step 3: Use vector addition from point P

To find relationships of \(\overrightarrow{PA}\), \(\overrightarrow{PB}\), and \(\overrightarrow{PC}\), use vector addition: \(\overrightarrow{PA} = \overrightarrow{A} - \overrightarrow{P}\), \(\overrightarrow{PB} = \overrightarrow{B} - \overrightarrow{P}\), and \(\overrightarrow{PC} = \overrightarrow{C} - \overrightarrow{P}\).

Step 4: Substitute midpoint vector into expression

Substitute \(\overrightarrow{C}\) from Step 2 into \(\overrightarrow{PC} = \overrightarrow{C} - \overrightarrow{P}\) to get \(\overrightarrow{PC} = \frac{1}{2} (\overrightarrow{A} + \overrightarrow{B}) - \overrightarrow{P}\).

Step 5: Calculate \(\overrightarrow{PA} + \overrightarrow{PB}\)

Sum \(\overrightarrow{PA}\) and \(\overrightarrow{PB}\): \(\overrightarrow{PA} + \overrightarrow{PB} = (\overrightarrow{A} - \overrightarrow{P}) + (\overrightarrow{B} - \overrightarrow{P}) = \overrightarrow{A} + \overrightarrow{B} - 2\overrightarrow{P}\).

Step 6: Relate \(\overrightarrow{PA} + \overrightarrow{PB}\) to \(2\overrightarrow{PC}\)

Compare \(\overrightarrow{PA} + \overrightarrow{PB} = \overrightarrow{A} + \overrightarrow{B} - 2\overrightarrow{P}\) with \(2\overrightarrow{PC}\). Since \(2\overrightarrow{PC} = 2\left(\frac{1}{2}(\overrightarrow{A} + \overrightarrow{B}) - \overrightarrow{P}\right) = \overrightarrow{A} + \overrightarrow{B} - 2\overrightarrow{P}\), the two are equal.

Key Concepts

Midpoint Theorem

The midpoint theorem is a fundamental concept in geometry and vector algebra. It states that the midpoint of a line segment divides the segment into two equal parts. This can be particularly helpful when analyzing relationships between points and vectors.
Imagine we have a line segment \(\overline{AB}\). If \(\mathrm{C}\) is the midpoint, then \(\mathrm{C}\) splits \(\overline{AB}\) so that \(\overline{AC} = \overline{CB}\). Taking this into vector terms, if \(\overrightarrow{A}\) and \(\overrightarrow{B}\) are position vectors of points \(A\) and \(B\) respectively, then the midpoint \(\overrightarrow{C}\) is expressed as: \\[\overrightarrow{C} = \frac{1}{2}(\overrightarrow{A} + \overrightarrow{B})\]
This formula is useful because it gives us a way to easily compute where the midpoint lies in the plane using vector coordinates. Whenever dealing with midpoints and vector relationships, remember this: the midpoint vector is the average of the two endpoints' vectors.

Vector Addition

Vector addition is a foundational operation in vector algebra that helps in combining vectors. Imagine placing a vector on the tip of another or adding two sides of a triangle. That's conceptually what vector addition does.
For vectors \(\overrightarrow{A}\) and \(\overrightarrow{B}\), vector addition is expressed as: \\[\overrightarrow{A} + \overrightarrow{B}\]
Using our exercise's context, if \(\mathrm{P}\) is any point outside \(\overline{AB}\), we can use point \(P\) to derive directional vectors like so: \(\overrightarrow{PA} = \overrightarrow{A} - \overrightarrow{P}\) and \(\overrightarrow{PB} = \overrightarrow{B} - \overrightarrow{P}\). When these directional vectors are summed, \(\overrightarrow{PA} + \overrightarrow{PB}\), it results in: \\[\overrightarrow{A} + \overrightarrow{B} - 2\overrightarrow{P}\]
Vector addition thus provides a tool to handle and simplify expressions involving multiple vectors and is particularly good for connecting points geometrically and algebraically.

Position Vector

A position vector is a useful concept when dealing with coordinates and vectors. It defines a vector that originates from a fixed reference point, usually the origin, to a particular point in space.
In coordinate geometry, if point \(A (x_1, y_1)\) has coordinates, its position vector would be: \\[\overrightarrow{A} = (x_1, y_1)\]
This vector forms a line from the origin \(O\) to \(A\). Position vectors are instrumental in expressing other vectors too. For example, \(\overrightarrow{P A}\) is the vector pointing from point \(P\) to \(A\). We calculate it using the positions of these points as \(\overrightarrow{PA} = \overrightarrow{A} - \overrightarrow{P}\).
Position vectors simplify the process of handling multiple vectors, as in determining the location of the midpoint or any other related vector.