Vectors In Two Dimensions (Copy)
VECTORS IN TWO DIMENSIONS — O LEVEL & IGCSE ADDITIONAL MATHEMATICS FORMULA SHEET + COMMON MISTAKES
| Topic | Formula / Rule | Common Mistake |
|---|---|---|
| Vector Notation | a = (x, y) | Mixing coordinates and vectors |
| Column Vector | (x, y)ᵀ | Writing as point only |
| Magnitude of Vector | |a| = √(x² + y²) | Forgetting square root |
| Equal Vectors | Same magnitude and direction | Using magnitude only |
| Negative Vector | −a = (−x, −y) | Negating one component only |
| Position Vector | Vector from origin | Mixing with displacement |
| Vector Addition | (a₁, a₂) + (b₁, b₂) = (a₁ + b₁, a₂ + b₂) | Adding wrongly |
| Vector Subtraction | a − b | Reversing order |
| Scalar Multiplication | k(x, y) = (kx, ky) | Multiplying one component only |
| Unit Vector | a/|a| | Forgetting division by magnitude |
| Direction Vector | Shows movement direction | Using position vector incorrectly |
| Parallel Vectors | One is scalar multiple of other | Using equal vectors condition |
| Collinear Points | Same direction vector | Using distance only |
| Distance Formula | √[(x₂ − x₁)² + (y₂ − y₁)²] | Missing square root |
| Midpoint Formula | ((x₁ + x₂)/2 , (y₁ + y₂)/2) | Dividing one coordinate only |
| Section Formula | Ratio division of line | Reversing ratio |
| Vector Equation of Line | r = a + λb | Mixing λ and coordinates |
| Parameter λ | Scalar multiplier | Treating as coordinate |
| Position Vector Addition | OA + AB = OB | Wrong direction |
| Triangle Law | a + b | Incorrect order |
| Polygon Law | Add vectors head-to-tail | Random arrangement |
| Resultant Vector | Sum of vectors | Subtracting instead |
| Magnitude Squared | |a|² = x² + y² | Taking root unnecessarily |
| Dot Product | a·b = x₁x₂ + y₁y₂ | Multiplying vectors directly |
| Dot Product Rule | a·b = |a||b|cosθ | Forgetting cosine |
| Perpendicular Vectors | a·b = 0 | Using equal magnitudes |
| Angle Between Vectors | cosθ = (a·b)/(|a||b|) | Wrong denominator |
| Parallel Condition | a = kb | Missing scalar multiple |
| Unit Direction Vector | a/|a| | Leaving unsimplified |
| Translation Vector | Describes movement | Wrong sign direction |
| Vector Ratio Problems | Use proportional vectors | Using midpoint formula |
| Vector Proof | Express in terms of vectors | Using coordinates only |
| Closed Polygon | Sum of vectors = 0 | Leaving extra vector |
| Vector Geometry | Use vector relationships | Using trigonometry unnecessarily |
| Bearing Vectors | Direction from north clockwise | Wrong orientation |
| Displacement Vector | Final − initial position | Reversing subtraction |
| Component Form | Horizontal and vertical parts | Swapping components |
| Scalar Quantity | Magnitude only | Giving direction |
| Vector Quantity | Magnitude and direction | Giving scalar only |
Written and Compiled By Sir Hunain Zia (AYLOTI), World Record Holder With 154 Total A Grades, 11 World Records and 7 Distinctions, Educate A Change.
QUICK VECTOR FORMULAS TABLE
| Formula | Expression |
|---|---|
| Magnitude | |a| = √(x² + y²) |
| Unit Vector | a/|a| |
| Distance Formula | √[(x₂ − x₁)² + (y₂ − y₁)²] |
| Midpoint Formula | ((x₁ + x₂)/2 , (y₁ + y₂)/2) |
| Dot Product | a·b = x₁x₂ + y₁y₂ |
| Angle Formula | cosθ = (a·b)/(|a||b|) |
QUICK VECTOR OPERATIONS TABLE
| Operation | Rule |
|---|---|
| Addition | Add corresponding components |
| Subtraction | Subtract corresponding components |
| Scalar Multiplication | Multiply each component |
| Negative Vector | Reverse all signs |
| Parallel Test | Scalar multiple |
| Perpendicular Test | Dot product = 0 |
Written and Compiled By Sir Hunain Zia (AYLOTI), World Record Holder With 154 Total A Grades, 11 World Records and 7 Distinctions, Educate A Change.
QUICK GEOMETRY RULES
| Situation | Rule |
|---|---|
| Collinear points | Parallel vectors |
| Midpoint | Average coordinates |
| Triangle vectors | a + b |
| Closed polygon | Vector sum = 0 |
| Position vector | From origin |
| Displacement vector | Final − initial |
QUICK VECTOR RELATIONSHIPS
| Relationship | Condition |
|---|---|
| Equal vectors | Same magnitude and direction |
| Parallel vectors | a = kb |
| Perpendicular vectors | a·b = 0 |
| Unit vector | Magnitude = 1 |
EXAM HACKS
| Situation | Fast Trick |
|---|---|
| Magnitude question | Use Pythagoras |
| Parallel vectors | Compare ratios |
| Perpendicular vectors | Dot product = 0 |
| Midpoint question | Average coordinates |
| Unit vector | Divide by magnitude |
| Translation | Final − initial |
| Geometry proof | Use vector equations |
| Angle between vectors | Use dot product |
Written and Compiled By Sir Hunain Zia (AYLOTI), World Record Holder With 154 Total A Grades, 11 World Records and 7 Distinctions, Educate A Change.
MOST COMMON LOSSES OF MARKS
| Mistake | Why Marks Are Lost |
|---|---|
| Reversing subtraction order | Wrong vector direction |
| Forgetting square root | Wrong magnitude |
| Using coordinates as vectors | Incorrect notation |
| Dot product errors | Wrong angle/perpendicular result |
| Missing scalar multiple | Parallel proof fails |
| Wrong midpoint calculation | Incorrect coordinates |
| Arithmetic sign mistakes | Entire vector incorrect |
| Forgetting unit vector simplification | Incomplete answer |
| Confusing scalar and vector quantities | Conceptual errors |
| Poor vector notation | Method marks lost |
Written and Compiled By Sir Hunain Zia (AYLOTI), World Record Holder With 154 Total A Grades, 11 World Records and 7 Distinctions, Educate A Change.