Position Vectors

• A position vector is a vector that starts at the origin $O(0,0)$ and ends at a specific point $P(x,y)$. It is conventionally denoted as $\vec{OP}$ or by a lowercase bold letter such as $\mathbf{p}$.

• The components of a position vector correspond exactly to the coordinates of the point. If point $A$ has coordinates $(x, y)$, its position vector is $\vec{OA} = x\mathbf{i} + y\mathbf{j}$ or in column form $\begin{pmatrix} x \\ y \end{pmatrix}$.

• The origin acts as the universal reference point. Every position vector is defined by its relationship to this single, fixed location, which distinguishes it from free vectors that can be translated anywhere in space.

• The relationship between position vectors and displacement vectors is governed by the Triangle Law of Vector Addition. To travel from point $A$ to point $B$, one can travel from $A$ back to the origin $O$ and then from $O$ to $B$.

• Mathematically, this is expressed as $\vec{AB} = \vec{AO} + \vec{OB}$. Since $\vec{AO}$ is the negative of the position vector $\vec{OA}$, the formula simplifies to the subtraction of position vectors.

• This principle allows us to calculate the vector between any two points in space simply by knowing their coordinates relative to the origin: $\vec{AB} = \mathbf{b} - \mathbf{a}$.

Calculating Displacement from Position

• To find the vector $\vec{AB}$ between two points $A(x_1, y_1)$ and $B(x_2, y_2)$, subtract the starting position vector from the ending position vector: $\vec{AB} = \begin{pmatrix} x_2 \\ y_2 \end{pmatrix} - \begin{pmatrix} x_1 \\ y_1 \end{pmatrix}$.

Finding Distance Between Points

• The distance between two points is the magnitude of their displacement vector. This is calculated using the Pythagorean theorem: $|\vec{AB}| = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

Dividing Line Segments

• To find a point $P$ that divides the line $AB$ in a ratio $\lambda : \mu$, use the formula: $\vec{OP} = \vec{OA} + \frac{\lambda}{\lambda + \mu}\vec{AB}$

• It is vital to distinguish between a point's location and the movement between points.

• The 'End minus Start' Rule: Always remember that $\vec{AB} = \text{Position of } B - \text{Position of } A$. Reversing this order is the most common source of sign errors in exams.

• Check the Origin: If a problem asks for a 'position vector', your final answer must represent a path starting from $O$. If you find a displacement vector, you may need to add it to a known position vector to reach the origin.

• Sanity Check with Sketches: Briefly sketch the points on a coordinate axis. If your vector $\vec{AB}$ has a positive $x$-component but point $B$ is to the left of point $A$, you have likely swapped the subtraction order.

• Unit Vectors: If asked for a unit vector in the direction of a position vector, divide the vector by its own magnitude: $\mathbf{\hat{a}} = \frac{\mathbf{a}}{|\mathbf{a}|}$.