Two- and Three-dimensional Vectors
Represent directed movement, combine component vectors and calculate two- or three-dimensional magnitude.
How to study National 5 Mathematics
Secure one operational skill at a time, connect it to neighbouring techniques, then apply it in unfamiliar contexts with complete working and a checked conclusion.
Core concepts
Concept 1
A vector has magnitude and direction and can be represented by a directed segment or components.
Exam cue: Write the route direction explicitly before combining named vectors.
Concept 2
Vectors add or subtract component by component; geometric pathways must share the same start and finish as the vector sought.
Exam cue: Align components vertically and protect negative signs.
Concept 3
Magnitude follows Pythagoras: square the components, add them and take the square root.
Exam cue: Check magnitude is non-negative and has the correct dimension.
Risk pitfalls and guardrails
Treating a vector as an undirected length.
Guardrail: Define directions and reference angles before beginning component or trigonometric work.
Reversing a directed segment without changing its sign.
Guardrail: Check signs, brackets, restrictions, units, accuracy and whether the answer needs a reason.
Adding components before squaring when finding magnitude.
Guardrail: Check signs, brackets, restrictions, units, accuracy and whether the answer needs a reason.
Memory anchors
Reverse vector
Reverse the direction and multiply by −1.
Vector addition
Add corresponding components.
Vector subtraction
Subtract corresponding components or add the negative vector.
Pathway rule
A route from A to C can be written as A to B plus B to C.
2D magnitude
√(x² + y²).
3D magnitude
√(x² + y² + z²).
Checkpoint rule
Do the check-up only after you can summarize each concept in one sentence and identify one dangerous pitfall from memory.
Knowledge Check (after reading)
Short check-up to confirm understanding of this module.
Check-up Questions
Add the vectors (3, −2) and (5, 4).
Calculate (7, 1) − (2, −5).
Answer all questions to submit.
Next step personalized recommendations
Continue learning
Move forward only after this module is stable.
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