Q.State True or False: Position vector of a point P is a vector whose initial point is origin.

Concept: Vector Equality — a vector is defined by its magnitude and direction, not by its location. The position vector of a point is defined as the vector from the origin to that point.

1. By definition, the position vector of a point $P$ is $\vec{OP}$, where $O$ is the origin.
2. This vector has its initial point fixed at $O$ and its terminal point at $P$. …

Why This Definition Matters

The idea of a position vector is one of the cleanest bridges between geometry and algebra. When you fix an origin $O$, every point $P$ in the plane or space gets a unique address: the vector $\overrightarrow{OP}$. This vector tells you exactly where the point is relative to the starting point $O$.

Without this convention, vectors would be free to slide around — a vector $\vec{v}$ could start anywhere. But a position vector pins the tail at the origin, so the coordinates of the point $P$ become the components of the vector. That’s why we can write $P = (x, y, z)$ and also $\overrightarrow{OP} = x\hat{i} + y\hat{j} + z\hat{k}$ interchangeably.

Step-by-Step Reasoning

1. Recall the formal definition.

   In coordinate geometry and vector algebra, the position vector of a point $P$ with respect to an origin $O$ is defined as the vector $\vec{r}_P = \overrightarrow{OP}$. The initial point is always $O$, and the terminal point is $P$.

2. Check the given statement.

   The statement says: “Position vector of a point P is a vector whose initial point is origin.”

   This matches the definition exactly. There is no ambiguity — the origin is the fixed reference, and the vector starts there.

3. Consider the alternative (what if it were false?).

   If the initial point were not the origin, you would have a free vector or a directed line segment between two arbitrary points, not a position vector. For example, $\overrightarrow{AB}$ is not a position vector unless $A$ happens to be the origin.

4. Confirm with a standard textbook example. …

Method: Verifying a Definition-Based True/False Statement

Use this when a statement simply restates (or subtly alters) a standard definition — here, the definition of a position vector.

Steps

Step 1: Recall the exact definition

The position vector of a point $P$ is $\overrightarrow{OP}$, drawn from the fixed origin $O$ to $P$; its initial point is, by definition, the origin.

Step 2: Compare the statement clause-by-clause with the definition

Check the claimed initial point (should be $O$) and terminal point (should be $P$). A definitional statement is true only if it agrees on every clause. …