Animated Solution for Mathematics - Vector Algebra: If $C$ is the mid point of $AB$ and $P$ is any point outside $AB$, then

Unlocking the Midpoint Identity

Welcome, fellow traveler on the path to JEE mastery. Today, we are not just solving a problem; we are uncovering a fundamental truth about the space around us.

We are looking at a simple line segment $AB$ with a midpoint $C$, and an external point $P$. It seems simple, almost trivial, but within this configuration lies the elegant power of vector algebra.

The Midpoint as an Average

Let us begin by grounding ourselves in the geometry. We have a segment $AB$ where point $C$ is the midpoint.

In the language of vectors, if we define the position vectors of $A$ and $B$ as $\vec{a}$ and $\vec{b}$ relative to some origin, then the position vector of the midpoint $C$, denoted as $\vec{c}$, is the arithmetic mean of the two endpoints:

Think of this as a balance point. If you placed equal weights at $A$ and $B$, the center of mass would be at $C$.

To make our algebra cleaner, let us multiply by $2$ to obtain the relation:

Keep this in your mental toolkit; it is the key that will unlock the final door.

The Bridge of Vectors

Now, we introduce the external point $P$. This point acts as our observer. We want to understand the vectors $\overrightarrow{PA}$, $\overrightarrow{PB}$, and $\overrightarrow{PC}$.

Remember the golden rule of vectors: the vector from point $X$ to point $Y$ is always the position vector of the destination minus the position vector of the source. Thus, we define:

I know it might feel like we are just shuffling symbols around, but look closely at what we have created. We have translated the geometric concept of 'distance and direction' into pure algebraic expressions. This is the transition from 'seeing' to 'calculating.'

The Algebraic Synthesis

Now, let us perform the operation the problem asks for: the sum $\overrightarrow{PA}+\overrightarrow{PB}$. Substituting our definitions, we get:

Grouping the terms, we see:

This is the moment of truth. Remember our midpoint relation from Phase 1, where $\vec{a}+\vec{b}=2\vec{c}$. Let us substitute that into our equation:

Factor out the $2$, and we get:

Since the expression $(\vec{c}-\vec{p})$ is exactly our definition for $\overrightarrow{PC}$, we have arrived at the final result.

The Elegance of Cancellation

We have arrived at the beautiful result:

Notice how the position vector $\vec{p}$ of our arbitrary point $P$ effectively 'disappears' from the relationship between the vectors. This confirms that the property is intrinsic to the geometry of the midpoint, independent of where we choose to place our observer $P$.

This is the power of vector algebra—it strips away the unnecessary details to reveal the underlying geometric skeleton. Keep this result in your arsenal; it is a powerful shortcut for many vector problems you will face in your JEE journey.