Vectors PM2
📐 VECTORS: Complete Study Guide
Cambridge AS & A Level Mathematics - Pure Mathematics 3
1. INTRODUCTION TO VECTORS
What is a Vector?
A vector is a quantity that has two essential properties:
- 📏 Magnitude (Size/Length)
- 🧭 Direction
Scalars vs Vectors
| Scalars (Number only) | Vectors (Number + Direction) |
|---|---|
| Temperature (25°C) | Velocity (60 km/h North) |
| Mass (5 kg) | Force (10 N Downward) |
| Distance (10m) | Displacement (10m East) |
2. DISPLACEMENT VECTORS (Section 9.1)
2.1 Writing Vectors
Method 1: Column Vector
⎛ x ⎞
v = ⎜ y ⎟
⎝ z ⎠
Where x is movement along the x-axis, y along the y-axis, and z along the z-axis.
Visualizing a 2D Vector
2.2 Vector Arithmetic
Addition & Subtraction
Simply add or subtract the corresponding top, middle, and bottom numbers.
⎛ 2 ⎞ ⎛ 5 ⎞ ⎛ 7 ⎞
⎜ 3 ⎟ + ⎜ 1 ⎟ = ⎜ 4 ⎟
⎝ 4 ⎠ ⎝ 2 ⎠ ⎝ 6 ⎠
Geometry: AB + BC = AC (Triangle Law)
2.3 Magnitude (Length)
Use Pythagoras theorem in 3D:
|v| = √(x² + y² + z²)
2.4 Unit Vectors
A unit vector (â) has a length of 1.
vec(a)
â = --------
|vec(a)|
3. POSITION VECTORS (Section 9.2)
A Position Vector starts from the Origin (O).
If Point A is (2, 3, 5), then position vector OA = 2i + 3j + 5k.
Finding Vector between 2 Points
To go from A to B:
AB = OB - OA
"Destination minus Start"
4. THE SCALAR PRODUCT (Section 9.3)
Also called the Dot Product. The result is a number, not a vector.
Formula 1 (Components)
a · b = (x₁x₂) + (y₁y₂) + (z₁z₂)
Formula 2 (Geometric)
a · b = |a| × |b| × cos(θ)
⚡ Special Cases:
- Perpendicular: Dot Product = 0 (cos 90° = 0)
- Parallel: Dot Product = |a||b|
4.4 Finding Angles Between Vectors
Rearrange the formulas to find θ:
a · b
cos θ = -----------
|a| × |b|
Example: Find angle between a = (1, 2, 2) and b = (2, 0, -1).
1. Dot Product: (1)(2) + (2)(0) + (2)(-1) = 2 + 0 - 2 = 0
2. Conclusion: Since dot product is 0, cos θ = 0.
3. Angle: θ = 90° (Perpendicular).
1. Dot Product: (1)(2) + (2)(0) + (2)(-1) = 2 + 0 - 2 = 0
2. Conclusion: Since dot product is 0, cos θ = 0.
3. Angle: θ = 90° (Perpendicular).
5. VECTOR EQUATIONS OF LINES (Section 9.4)
The Formula
A line is defined by a fixed point (a) and a direction (b).
r = a + tb
- r: General point on the line (x, y, z)
- a: Position vector of a known point
- b: Direction vector
- t: Scalar parameter
Parametric Form
Splitting the vector equation into coordinates:
x = a₁ + t(b₁)
y = a₂ + t(b₂)
z = a₃ + t(b₃)
6. INTERSECTION OF LINES (Section 9.5)
Three Possibilities in 3D
PARALLEL INTERSECTING SKEW
// X /
// / \ -----
// / \ /
(Never meet) (One point) (Miss each other)
Step-by-Step Solving
- Check Directions: Are direction vectors multiples? If yes -> Parallel.
- Equate Coordinates: Set x = x, y = y to form simultaneous equations.
- Solve: Find the parameters (e.g., s and t).
- Verify: Plug s and t into the z-equation.
The "Skew" Test:
If the values of s and t work for x and y, but FAIL the z-equation (e.g., 5 = 9), the lines are SKEW. They do not intersect.
If the values of s and t work for x and y, but FAIL the z-equation (e.g., 5 = 9), the lines are SKEW. They do not intersect.
7. SUMMARY & CHEAT SHEET
| Concept | Formula |
|---|---|
| Magnitude | √(x² + y² + z²) |
| Unit Vector | vec(v) ÷ |v| |
| Scalar Product | x₁x₂ + y₁y₂ + z₁z₂ |
| Angle | cos⁻¹( (a·b) / (|a||b|) ) |
| Line Eq | r = a + tb |
| Perpendicular | a · b = 0 |
🎓 Exam Tip: When finding the angle between two lines, always use their direction vectors, not their position vectors!
