Position & Displacement Vectors

• Position vector: A position vector gives the location of a point relative to an origin, so it is tied to a specific reference point. In 2D, its components match the point coordinates, which is why location and vector notation are interchangeable in coordinate geometry. This idea applies whenever you need to encode a point as a vector object.

• Displacement vector: A displacement vector describes the directed movement from one point to another, not the absolute location of either point. It captures both direction and magnitude of change, so reversing endpoints changes the sign. This is the correct object for motion, translation, and relative position questions.

• Notation and direction: If points have position vectors $\mathbf{a}$ and $\mathbf{b}$, then the vector from the first point to the second is order-sensitive. Swapping the order gives the opposite vector, which preserves the same length but reverses direction. Direction sensitivity is the main conceptual difference between vector difference and scalar distance.

• Difference of positions gives displacement: The identity $\vec{AB}=\vec{OB}-\vec{OA}$ follows from path addition, because going from $A$ to $B$ is equivalent to going $A \to O$ then $O \to B$. This is why subtraction appears naturally in displacement problems. It applies in any dimension as long as vectors are expressed in the same basis and origin.

• Component-wise structure: If $\mathbf{a}=(x_1,y_1)$ and $\mathbf{b}=(x_2,y_2)$, then $\vec{AB}=(x_2-x_1,\,y_2-y_1)$. Each component measures signed change along one axis, so negative values are meaningful directional information rather than errors. This component view links vector algebra directly to coordinate geometry.

• Key relations to memorize with meaning: > Core formulas: $\vec{AB}=\mathbf{b}-\mathbf{a}$ and $|\vec{AB}|=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$. The first formula gives direction and relative movement, while the second strips direction to keep only length. Use the first for vector equations and the second for distances.

• Method 1: Vector subtraction workflow: Start by naming position vectors clearly, for example $\vec{OA}=\mathbf{a}$ and $\vec{OB}=\mathbf{b}$. Compute displacement as $\vec{AB}=\mathbf{b}-\mathbf{a}$ by subtracting corresponding components. Finish by checking whether the sign pattern matches the expected direction on a sketch.

• Method 2: Coordinate-difference workflow: Convert position vectors to coordinates first, then apply $(\Delta x,\Delta y)=(x_2-x_1,\,y_2-y_1)$. This route is fast when points are given in coordinate form and reduces notation overhead. It is equivalent to Method 1, so both should produce identical components.

• Decision criterion and validation: Use vector notation when chaining paths or proving relationships, and use coordinate form when asked for distance or direct numerical displacement. In either method, verify by reverse-direction logic: $\vec{BA}=-\vec{AB}$. A quick reverse check catches most sign-order mistakes.

• Conceptual contrasts matter for marks: Students lose accuracy when they mix location with movement, or vector quantity with scalar quantity. Treat each object by what it represents before calculating. This prevents formula misuse in multi-step questions.

Comparison table

• Second distinction: Distance $|\vec{AB}|$ is scalar, while $\vec{AB}$ is vector. Use magnitude only when direction is irrelevant, such as pure length requests. Keep full vector form when direction or further vector operations are required.

• Always label point order first: Write a mini statement like "from $A$ to $B$" before any algebra. This locks in $\vec{AB}=\mathbf{b}-\mathbf{a}$ and avoids the most common sign inversion. Examiners reward consistent direction logic across all parts.

• Build a rapid self-check routine: After computing, test one sanity condition such as $\vec{AB}+\vec{BA}=\mathbf{0}$. If this fails, the error is usually order or bracket handling in subtraction. A 10-second check often saves full-method marks.

• Use structure in multi-part questions: Keep symbolic vectors as long as possible, then substitute values late to reduce arithmetic slips. If a magnitude is requested, separate the stages: first find displacement components, then apply the square-root formula. This separation prevents mixing vector and scalar steps.

• Mistaking $\mathbf{a}-\mathbf{b}$ for $\vec{AB}$: The displacement from $A$ to $B$ is $\mathbf{b}-\mathbf{a}$, not the reverse. Reversing order gives the opposite direction, which may still look numerically plausible and therefore dangerous. Always map subtraction to endpoint minus start point.

• Confusing vector with magnitude: Writing $\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$ when the question asks for displacement drops direction information. This turns a vector answer into a scalar and changes the object being solved for. Read command words such as "vector", "displacement", and "distance" carefully.

• Ignoring common reference setup: Position vectors must be interpreted from the same origin and coordinate basis before subtraction. If coordinate systems or origins differ, raw subtraction is invalid until converted. This misconception causes hidden conceptual errors even when arithmetic is correct.