Magnitude of A Vector

7.3 Magnitude of a vector – Cheat Sheet

1. Definition

• The magnitude of a vector is its length.
• For a vector

[ x ]
[ y ]

the magnitude is given by:

|v| = √(x² + y²)

2. Notation

• |a| = magnitude of vector a
• |AB| = magnitude of vector from point A to point B

3. Example 1 – Find the magnitude of

[ 3 ]
[ 4 ]

Solution:

|v| = √(3² + 4²)  
     = √(9 + 16)  
     = √25  
     = 5

4. Example 2 – Magnitude of a displacement vector
If AB =

[ -5 ]
[ 12 ]

then:

|AB| = √((-5)² + (12)²)  
      = √(25 + 144)  
      = √169  
      = 13

5. Key points to remember

• Magnitude is always positive.
• Works for all 2D vectors, including negative components.
• Formula comes from Pythagoras’ theorem.