MathematicsChapter 7: Vectors in Plane
Magnitude
|xi + yj| = sqrt(x^2 + y^2), Pythagoras.
Learning objectives
- Explain magnitude using the correct definitions, conditions, and notation.
- Apply sqrt(x^2+y^2), never negative correctly in routine and exam-style questions.
- Show each important step, simplify the result, and check that it satisfies the original conditions.
Definition
|xi + yj| = sqrt(x^2 + y^2), Pythagoras. |3i-4j|=5. |-5i+12j|=13. |0|=0. Magnitude is never negative. Equal vectors have equal magnitudes, but equal magnitudes do not mean equal vectors (directions may differ).
Formula
u = 3i - 4j. Find
Where / symbols
- MAGNITUDE
- |xi + yj| = sqrt(x^2 + y^2), Pythagoras. |3i-4j|=5. |-5i+12j|=13. |0|=0. Magnitude is never negative. Equal vectors have equal magnitudes, but equal magnitudes do not mean equal vectors (directions may differ).
- COMPUTE
- Square, add, square root. 3-4-5 and 5-12-13 triangles are the usual integers. Leave sqrt(8)=2 sqrt(2) simplified if asked.
- PROPERTIES
- |-u|=|u|. |u|=0 iff u is the zero vector. Distance from A to B is |AB|.
Overview
Pythagoras on components:
Compute
Square, add, square root. 3-4-5 and 5-12-13 triangles are the usual integers. Leave sqrt(8)=2 sqrt(2) simplified if asked.
Examples |3i-4j|=5 · |-5i+12j|=13 · |i+j|=sqrt(2)
Properties
|-u|=|u|. |u|=0 iff u is the zero vector. Distance from A to B is |AB|.
Examples |-u|=|u| · |OP| is distance OP · Not x+y
WHAT — MAGNITUDE
|xi + yj| = sqrt(x^2 + y^2), Pythagoras. |3i-4j|=5. |-5i+12j|=13. |0|=0. Magnitude is never negative. Equal vectors have equal magnitudes, but equal magnitudes do not mean equal vectors (directions may differ). The symbols and conditions are part of the mathematics, so include them when stating the rule.
WHY — Why this topic matters
Magnitude matters because vectors describe both size and direction, so they are useful for displacement, velocity, force, and geometry. Components turn a directional problem into manageable algebra.
HOW — How to solve and explain it
Start by identifying what is given and what must be found. Work through COMPUTE, then PROPERTIES. Worked: u = 3i - 4j. Find |u|. |u|^2 = 3^2 + (-4)^2 |u|^2 = 9 + 16 |u|^2 = 25 |u| = 5 Finish by checking signs, restrictions, units, and whether the answer matches the question.
Worked example
- u = 3i - 4j. Find |u|.
- |u|2 = 32 + (-4)2
- |u|2 = 9 + 16
- |u|2 = 25
- |u| = 5
In simple words
|3i-4j| is not |3-4|=1. You square the components, not subtract them first.
Key points
- sqrt(x^2+y^2)
- never negative
- squares then sqrt
- zero iff both components 0
Exam preparation
Exam Focus
Quick revision
Revision Zone
Review the key facts, formulas and checks before you move on.Magnitude
Sticky Notes · Quick Revision
Core Idea
- |xi + yj| = sqrt(x^2 + y^2), Pythagoras. |3i-4j|=5. |-5i+12j|=13. |0|=0. Magnitude is never negative. Equal vectors have equal magnitudes, but equal magnitudes do not mean equal vectors (directions may differ).
COMPUTE
- Square, add, square root. 3-4-5 and 5-12-13 triangles are the usual integers. Leave sqrt(8)=2 sqrt(2) simplified if asked.
PROPERTIES
- |-u|=|u|. |u|=0 iff u is the zero vector. Distance from A to B is |AB|.
Worked Method
- Worked: u = 3i - 4j. Find |u|.
- |u|2 = 32 + (-4)2
- |u|2 = 9 + 16
- |u|2 = 25
- |u| = 5
Common Misconception
- |3i-4j| is not |3-4|=1. You square the components, not subtract them first.
FBISE Exam Check
- |3i-4j| is not |3-4|=1. You square the components, not subtract them first.
60-second revision
- |xi + yj| = sqrt(x2 + y2), Pythagoras.
- COMPUTE: Square, add, square root.
- PROPERTIES: |-u|=|u|.
- Worked-method reminder: |u| = 5
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