11-13-2025, 01:38 PM
Vectors — Beginner Reference Sheet (GCSE & A-Level Friendly)
A clear, simple introduction to vectors — perfect for GCSE, A-Level, physics, computer science, and challenge problems.
-----------------------------------------------------------------------
1. What Is a Vector?
A vector has:
• a magnitude (size)
• a direction
Examples:
• displacement
• velocity
• force
• acceleration
Vectors are written as:
• arrows → →
• bold letters → a, v
• column vectors → [x; y]
-----------------------------------------------------------------------
2. Column Vector Notation
A 2D vector is written as:
[ x ]
[ y ]
Where:
• x = movement left/right
• y = movement up/down
Example:
[ 3 ]
[ -2 ] means 3 right, 2 down.
-----------------------------------------------------------------------
3. Adding & Subtracting Vectors
Add or subtract components individually.
Example:
Another example:
-----------------------------------------------------------------------
4. Multiplying Vectors by Scalars
Multiply each component by the number.
Example:
2 ×
[ 3 ]
[ -1 ]
=
[ 6 ]
[ -2 ]
Negative scalars change direction.
-----------------------------------------------------------------------
5. Magnitude (Length) of a Vector
Use Pythagoras:
|v| = √(x² + y²)
Example:
v = [3; 4]
|v| = √(3² + 4²) = √25 = 5
This is essential in physics and algebra.
-----------------------------------------------------------------------
6. Direction of a Vector
Direction angle θ measured from the x-axis:
θ = tan⁻¹(y / x)
Example:
v = [3; 3]
θ = tan⁻¹(3/3) = tan⁻¹(1) = 45°
-----------------------------------------------------------------------
7. Unit Vectors
A unit vector has length 1.
To convert any vector into a unit vector:
unit v = v / |v|
Example:
v = [4; 0] → magnitude = 4
unit vector = [1; 0]
-----------------------------------------------------------------------
8. Parallel & Perpendicular Vectors
Parallel:
Two vectors are parallel if one is a scalar multiple of the other.
Example:
[2; 4] is parallel to [1; 2]
Perpendicular:
x₁x₂ + y₁y₂ = 0 (their dot product is zero)
Example:
[2; 1] and [-1; 2]
(2)(-1) + (1)(2) = -2 + 2 = 0 → perpendicular.
-----------------------------------------------------------------------
9. Resultant Vectors (Physics)
Two forces acting together = vector addition.
Example:
Force 1: [3; 4]
Force 2: [1; -2]
Resultant = [4; 2]
Magnitude of resultant:
|R| = √(4² + 2²) = √20 = 4.47 N
-----------------------------------------------------------------------
10. Common Mistakes
❌ Adding magnitudes instead of components
❌ Forgetting negatives
❌ Wrong square root in magnitude
❌ Mixing row and column notation
❌ Using tan(x/y) instead of tan(y/x)
✔ Always:
• add x’s together
• add y’s together
• use √(x² + y²) for magnitude
• use tan⁻¹(y/x) for angle
-----------------------------------------------------------------------
Summary
Key rules to remember:
• Vectors = magnitude + direction
• Add/subtract components
• Multiply by scalars component-wise
• Magnitude = √(x² + y²)
• Angle = tan⁻¹(y/x)
• Parallel = multiples
• Perpendicular = dot product = 0
Master these and you'll handle every GCSE/A-Level vector question easily.
A clear, simple introduction to vectors — perfect for GCSE, A-Level, physics, computer science, and challenge problems.
-----------------------------------------------------------------------
1. What Is a Vector?
A vector has:
• a magnitude (size)
• a direction
Examples:
• displacement
• velocity
• force
• acceleration
Vectors are written as:
• arrows → →
• bold letters → a, v
• column vectors → [x; y]
-----------------------------------------------------------------------
2. Column Vector Notation
A 2D vector is written as:
[ x ]
[ y ]
Where:
• x = movement left/right
• y = movement up/down
Example:
[ 3 ]
[ -2 ] means 3 right, 2 down.
-----------------------------------------------------------------------
3. Adding & Subtracting Vectors
Add or subtract components individually.
Example:
Code:
[3] [1] [4]
[2] + [5] = [7]Another example:
Code:
[6] [2] [4]
[4] - [1] = [3]-----------------------------------------------------------------------
4. Multiplying Vectors by Scalars
Multiply each component by the number.
Example:
2 ×
[ 3 ]
[ -1 ]
=
[ 6 ]
[ -2 ]
Negative scalars change direction.
-----------------------------------------------------------------------
5. Magnitude (Length) of a Vector
Use Pythagoras:
|v| = √(x² + y²)
Example:
v = [3; 4]
|v| = √(3² + 4²) = √25 = 5
This is essential in physics and algebra.
-----------------------------------------------------------------------
6. Direction of a Vector
Direction angle θ measured from the x-axis:
θ = tan⁻¹(y / x)
Example:
v = [3; 3]
θ = tan⁻¹(3/3) = tan⁻¹(1) = 45°
-----------------------------------------------------------------------
7. Unit Vectors
A unit vector has length 1.
To convert any vector into a unit vector:
unit v = v / |v|
Example:
v = [4; 0] → magnitude = 4
unit vector = [1; 0]
-----------------------------------------------------------------------
8. Parallel & Perpendicular Vectors
Parallel:
Two vectors are parallel if one is a scalar multiple of the other.
Example:
[2; 4] is parallel to [1; 2]
Perpendicular:
x₁x₂ + y₁y₂ = 0 (their dot product is zero)
Example:
[2; 1] and [-1; 2]
(2)(-1) + (1)(2) = -2 + 2 = 0 → perpendicular.
-----------------------------------------------------------------------
9. Resultant Vectors (Physics)
Two forces acting together = vector addition.
Example:
Force 1: [3; 4]
Force 2: [1; -2]
Resultant = [4; 2]
Magnitude of resultant:
|R| = √(4² + 2²) = √20 = 4.47 N
-----------------------------------------------------------------------
10. Common Mistakes
❌ Adding magnitudes instead of components
❌ Forgetting negatives
❌ Wrong square root in magnitude
❌ Mixing row and column notation
❌ Using tan(x/y) instead of tan(y/x)
✔ Always:
• add x’s together
• add y’s together
• use √(x² + y²) for magnitude
• use tan⁻¹(y/x) for angle
-----------------------------------------------------------------------
Summary
Key rules to remember:
• Vectors = magnitude + direction
• Add/subtract components
• Multiply by scalars component-wise
• Magnitude = √(x² + y²)
• Angle = tan⁻¹(y/x)
• Parallel = multiples
• Perpendicular = dot product = 0
Master these and you'll handle every GCSE/A-Level vector question easily.