Magnitude

Magnitude

What is it?

|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).

In simple words

• |xi + yj| = sqrt(x² + y²), Pythagoras.
• COMPUTE: Square, add, square root.
• PROPERTIES: |-u|=|u|.
• Worked-method reminder: |u| = 5

Important terms

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|.

What you should learn

• 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.

Examples

• |3i-4j|=5
• |-5i+12j|=13
• |i+j|=sqrt(2)

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|.

Full school-level explanation

|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).

Pythagoras on components:

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.

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.

Remember

• squares then sqrt

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|.

Remember

• zero iff both components 0

Examples

• |-u|=|u|
• |OP| is distance OP
• Not x+y

Remember

• sqrt(x^2+y^2)
• never negative
• |u|=5
• squares of components
• squares then sqrt
• zero iff both components 0
• sqrt(x^2+y^2)
• never negative

How the ideas connect

Worked: u = 3i - 4j. Find |u|. |u|² = 3² + (-4)² |u|² = 9 + 16 |u|² = 25 |u| = 5

Deeper understanding

Go beyond memorising Magnitude. Connect the definition to reasons, relationships, exceptions, and exam wording.

How the ideas connect

Worked: u = 3i - 4j. Find |u|. |u|² = 3² + (-4)² |u|² = 9 + 16 |u|² = 25 |u| = 5

Common mistakes

Exam focus

Core Idea

COMPUTE

PROPERTIES

Worked Method

Exam focus

Quick revision

• sqrt(x^2+y^2)
• never negative
• |u|=5
• squares of components
• MAGNITUDE: |xi + yj| = sqrt(x^2 + y^2), Pythagoras. |3i-4j|=5. |-5i+12j|=13. |0|=0. Magnitude is never negative
• COMPUTE: Square, add, square root. 3-4-5 and 5-12-13 triangles are the usual integers. Leave sqrt(8)=2 sqrt(2

|3i-4j| is not |3-4|=1. You square the components, not subtract them first.

For long answers on Magnitude: define clearly → explain the main parts → give one example → state one exam caution.