Position vectors locate points relative to a fixed origin, while displacement vectors describe movement between two points regardless of where the origin is drawn. The core relation AB=b−a links these ideas and turns geometry into component arithmetic. Mastery comes from consistent direction labeling, endpoint order, and quick checks that signs and magnitudes match the diagram.
1. Definition & Core Concepts
Coordinate grid showing origin O, points A and B, position vectors OA and OB, and displacement vector AB illustrating AB equals b minus a.
2. Underlying Principles
3. Methods & Techniques
4. Key Distinctions
5. Exam Strategy & Tips
6. Common Pitfalls & Misconceptions
What each vector means
Position vector: A position vector gives the location of a point measured from a chosen origin, so it encodes "where" the point is in space. If point A has coordinates (x,y), then a=OA=(xy) because the components are exactly the coordinate differences from the origin. This is useful whenever geometry is rewritten as algebra.
Displacement vector: A displacement vector gives the directed change from one point to another, so it encodes "how to get from start to finish." It depends on both start and end points, not just one point. This makes it ideal for describing movement, translation, and relative position.
Direction and notation
Directed segment notation: AB means start at A and end at B, so reversing order changes the sign. That is why BA=−AB, even though both have the same magnitude. Direction is part of the object, not an optional label.
Coordinate interpretation: In component form, horizontal and vertical changes are tracked separately and then combined into one column vector. This separates geometry into two one-dimensional differences, which reduces sign mistakes. The idea extends directly to 3D by adding a third component.
Why subtraction appears
Triangle law of vectors: Going from A to B can be decomposed as going from A to O and then from O to B. Algebraically, this is AB=AO+OB=−a+b. This principle works because vector addition tracks net directed change.
Memorize the key identity: > Key formula: AB=b−a where a=OA and b=OB. The endpoint vector comes first and the start-point vector is subtracted, which reflects final minus initial state. This is the same logic used in physics for change in position.
Invariance and structure
Origin choice and consistency: A position vector depends on where origin O is chosen, but a displacement between two fixed points does not change if the origin shifts. This matters because physical motion and geometric translation are relative differences, not absolute placements. So displacement is often the more robust quantity in modeling.
Componentwise reasoning: If A=(x1,y1) and B=(x2,y2), then AB=(x2−x1y2−y1). Each component is an independent signed difference, which explains why order reversal flips both signs. The method is reliable because it directly encodes horizontal and vertical change.
Standard workflow for finding displacement
Step 1: Define vectors clearly: Write position vectors with explicit labels, such as a=OA and b=OB. Clear labeling prevents endpoint confusion before algebra starts. This setup is especially important when multiple points are present.
Step 2: Apply final-minus-initial: Compute AB=b−a, then subtract components row by row. This converts geometry into a short arithmetic operation with signs preserved. It is the fastest exam-safe technique for coordinate and vector-form questions.
Verification and reverse use
Step 3: Sanity-check direction: Quickly test whether the signs make geometric sense, for example rightward movement should give positive x-change. A rough sketch is enough to detect reversed subtraction. This prevents losing marks from one sign error.
Step 4: Use the relation backward when needed: If AB and one position vector are known, rearrange to find the other, such as b=a+AB. This is useful in reconstruction problems where a point location is missing. Rearrangement works because vector equations obey ordinary algebraic balancing.
Concept comparisons
| Feature | Position Vector | Displacement Vector |
| --- | --- | --- |
| Meaning | Location from origin | Change from one point to another |
| Typical form | OA=a | AB=b−a |
| Depends on origin? | Yes | No |
| Changes if points are swapped? | Not applicable | Yes, sign reverses |
This distinction matters because many errors come from treating a location as if it were a movement.
Direction-sensitive distinctions
| Comparison | Correct idea | Common confusion |
| --- | --- | --- |
| AB vs BA | Same magnitude, opposite direction | Assuming they are equal |
| b−a vs a−b | First is A→B, second is B→A | Forgetting endpoint order |
| Coordinates vs vectors | Coordinates locate points | Vectors describe directed differences |
Use these contrasts to choose the correct expression before calculating, not after.
High-yield exam habits
Write start and end points above each vector: Marking "A→B" before computing forces correct order in b−a. This tiny notation habit prevents the most frequent sign reversal error. It is faster than reworking full arithmetic later.
Always sketch a minimal diagram: Even when no graph is required, a two-point sketch with arrows gives an immediate direction check. You can confirm whether each component should be positive or negative before finalizing. This is a strong accuracy filter under time pressure.
Quick validation checklist
Check with reverse relation: Verify that BA=−AB from your final components. If this fails, subtraction order or sign handling is wrong. This is a one-line consistency test worth routine use.
Check endpoint reconstruction: Confirm that a+AB=b. If reconstruction works, your displacement is coherent with both points. This method catches arithmetic slips that direction intuition alone may miss.
Frequent mistakes
Mixing point labels and vector symbols: Students sometimes subtract point names or coordinates without defining vectors first. This breaks logical structure and increases notation errors. Define a and b first so each subtraction has meaning.
Ignoring that subtraction is not commutative: Treating b−a as interchangeable with a−b flips direction and changes interpretation. The result may have correct magnitude but wrong physical or geometric meaning. Always tie subtraction order to start and finish points.
Misreading what is being asked
Confusing "position of B" with "displacement to B": A question about position asks for OB, while movement from A to B asks for AB. These are connected but not interchangeable. Misidentifying the target vector can invalidate an otherwise correct calculation.
Believing displacement needs the origin: Displacement depends only on the two endpoints, so changing origin does not change AB. Thinking otherwise leads to unnecessary diagram dependence. Remember: displacement is relative difference, not absolute placement.
