Vectors
“A vector is an arrow: 'how much' plus 'which way'.”
The formula
|v| = √(x² + y²)How to read it: a vector's magnitude is the hypotenuse of the right triangle built from its horizontal part x and vertical part y
- v
- — a vector — a quantity with both size and direction
- (x, y)
- — the vector's components (horizontal and vertical amounts)
- |v|
- — the magnitude (length) of the vector
The hook
Velocity, force, wind — some quantities need more than 'how much'; they need 'in which direction' too. That's a vector.
In plain words
A vector is an arrow carrying both size and direction. Moving x sideways and y up is written (x,y).
The intuition
A vector is a 'move', not a 'place'. No matter where it starts, an arrow going 3 right and 4 up is the same vector. Adding two vectors means sticking the second arrow onto the tip of the first — which is why you just add component by component.
How it's built
The magnitude of vector (x,y) is the distance from the origin to that point — the hypotenuse of a right triangle, so by Pythagoras √(x²+y²). Addition works componentwise: (x₁,y₁)+(x₂,y₂)=(x₁+x₂, y₁+y₂).
Example
The magnitude of v=(3,4) is √(3²+4²)=√(9+16)=√25=5. The arrow going 3 right and 4 up is exactly 5 long.
Common mistake
A vector's magnitude is NOT x+y. The size of (3,4) is 5, not 7 — it's a diagonal length, so you must square, add, then take the root.
Where it's used
Force, velocity, and acceleration in physics; character movement in games; navigation headings; 3D graphics — anywhere you handle 'directed quantities'.
Where it came from
The vector idea was refined in 19th-century physics (especially electromagnetism) to handle force and velocity; Hamilton and Gibbs settled today's notation.
Prerequisites
Quick check
What is the magnitude of the vector (3, 4)?
- 5✓
- 7
- 12
- 25
Practice
Find the magnitude of the vector (6, 8).
Answer: 10
- √(6² + 8²) = √(36 + 64)
- = √100 = 10
Takeaway: Magnitude squares the components, adds, then takes the root.
Find the x-component of the vector (1, 2) + (3, 4).
Answer: 4
- Add component by component
- x-component: 1 + 3 = 4
Takeaway: Vector addition happens component by component.
Find the magnitude of the vector (5, 0).
Answer: 5
- √(5² + 0²) = √25
- = 5
Takeaway: A vector along one axis has magnitude equal to that component.
Explain why (−3, 4) has the same magnitude as (3, 4).
Answer: undefined
- |(−3,4)| = √((−3)² + 4²) = √(9+16) = 5
- Squaring removes the sign
- Only the direction differs; the length is the same
Takeaway: Magnitude drops direction (sign) and measures length only.