Position & Displacement Vectors

• Position Vector: A position vector describes the location of a specific point in space relative to a fixed reference point, known as the origin (usually denoted as $O$). It is typically represented by a lowercase bold letter, such as $\mathbf{a}$, and points from the origin to the point, e.g., $\vec{OA} = \mathbf{a}$.

• Components of a Position Vector: The components of a position vector are identical to the coordinates of the point it describes. For instance, a point with coordinates $(x, y)$ has a position vector $\begin{pmatrix} x \\ y \end{pmatrix}$ from the origin.

• Displacement Vector: A displacement vector describes the direction and distance (magnitude) required to move from one point to another. Unlike a position vector, it does not necessarily originate from the fixed origin but rather connects two arbitrary points, say from point $A$ to point $B$, denoted as $\vec{AB}$.

• Vector Subtraction for Displacement: The fundamental principle for finding a displacement vector $\vec{AB}$ between two points $A$ and $B$ is to subtract the position vector of the starting point ($A$) from the position vector of the ending point ($B$). This can be visualized as traveling from $A$ to the origin ($O$) and then from the origin to $B$.

• Path Equivalence: Mathematically, this translates to $\vec{AB} = \vec{AO} + \vec{OB}$. Since $\vec{AO}$ is the negative of the position vector $\vec{OA}$ (i.e., $\vec{AO} = -\mathbf{a}$), and $\vec{OB}$ is the position vector $\mathbf{b}$, the formula becomes $\vec{AB} = -\mathbf{a} + \mathbf{b}$.

• Standard Formula: This is commonly written as $\vec{AB} = \mathbf{b} - \mathbf{a}$. This rule is universally applicable for finding the displacement vector between any two points whose position vectors are known relative to a common origin.

• Calculating Displacement from Position Vectors: To find the displacement vector $\vec{AB}$ given position vectors $\mathbf{a} = \begin{pmatrix} a_x \\ a_y \end{pmatrix}$ and $\mathbf{b} = \begin{pmatrix} b_x \\ b_y \end{pmatrix}$, simply perform vector subtraction component-wise:

$\vec{AB} = \mathbf{b} - \mathbf{a} = \begin{pmatrix} b_x \\ b_y \end{pmatrix} - \begin{pmatrix} a_x \\ a_y \end{pmatrix} = \begin{pmatrix} b_x - a_x \\ b_y - a_y \end{pmatrix}$

• Graphical Representation: When representing position vectors, always draw them starting from the origin $O$. For displacement vectors, draw an arrow from the starting point to the ending point. The length of the arrow represents the magnitude, and the arrowhead indicates the direction.

• Using Coordinates Directly: If points $A(x_A, y_A)$ and $B(x_B, y_B)$ are given, their position vectors are $\mathbf{a} = \begin{pmatrix} x_A \\ y_A \end{pmatrix}$ and $\mathbf{b} = \begin{pmatrix} x_B \\ y_B \end{pmatrix}$. The displacement vector $\vec{AB}$ can then be found by subtracting the coordinates: $\vec{AB} = \begin{pmatrix} x_B - x_A \\ y_B - y_A \end{pmatrix}$.

• Position Vector vs. Displacement Vector: While both are vectors, a position vector is always tied to a fixed origin and defines a point's absolute A displacement vector, however, describes a relative change in position between two points and is independent of the origin's specific location, as long as the position vectors of the two points are defined from the same origin.

• Directionality: The direction of a displacement vector is crucial. $\vec{AB}$ represents the path from $A$ to $B$, while $\vec{BA}$ represents the path from $B$ to $A$. These two vectors have the same magnitude but opposite directions, meaning $\vec{BA} = -\vec{AB}$.

• Origin's Role: The origin is fundamental for defining position vectors, as they are always measured from it. For displacement vectors, while their calculation relies on position vectors relative to an origin, the displacement itself describes the 'journey' between two points, irrespective of the origin's specific coordinates.

• Always Sketch Diagrams: Even if not explicitly asked, drawing a simple coordinate plane with the origin and points $A$, $B$ can help visualize the vectors and prevent errors, especially when dealing with negative components or complex paths.

• Remember the 'End Minus Start' Rule: For a displacement vector $\vec{PQ}$, always think of it as 'position vector of $Q$ minus position vector of $P$' (i.e., $\mathbf{q} - \mathbf{p}$). A common mistake is to subtract in the wrong order.

• Check for Consistency: Ensure all position vectors are defined relative to the same origin. If different origins are implied or given, convert them to a common origin before performing calculations.

• Simplify Components: After calculating a displacement vector, always simplify its components if possible, for example, $\begin{pmatrix} 6 \\ -12 \end{pmatrix}$ should be presented as such, not left as $\begin{pmatrix} 10-4 \\ 1-13 \end{pmatrix}$.

• Incorrect Subtraction Order: A frequent error is calculating $\mathbf{a} - \mathbf{b}$ instead of $\mathbf{b} - \mathbf{a}$ for $\vec{AB}$. This results in a vector with the correct magnitude but the opposite direction, leading to incorrect answers in subsequent calculations.

• Confusing Coordinates with Vectors: While position vector components are the same as coordinates, it's important to remember that a vector has direction and magnitude, whereas a coordinate pair simply denotes a point's This distinction becomes critical when performing vector operations.

• Ignoring the Origin: Students sometimes forget that position vectors are inherently defined relative to an origin. When asked to sketch or interpret position vectors, they should always be drawn starting from the origin $O$ to the point.

• Misinterpreting Negative Components: A negative component in a vector simply indicates movement in the negative direction along that axis (e.g., left for x, down for y). It does not imply a 'smaller' vector, only a different direction.

• Vector Paths: Position and displacement vectors are the building blocks for understanding more complex vector paths, where a journey might involve multiple segments (e.g., $\vec{AC} = \vec{AB} + \vec{BC}$).

• Relative Motion: In physics, displacement vectors are crucial for describing relative positions and velocities between objects. For example, the position of object B relative to object A is given by $\mathbf{r}_{B/A} = \mathbf{r}_B - \mathbf{r}_A$, which is a displacement vector.

• Geometric Proofs: These vectors are extensively used in vector geometry to prove properties of geometric figures, such as collinearity of points or parallelism of lines, by showing that one vector is a scalar multiple of another.