Vectors | Grade 10
➡️ What is a Vector?
A vector is a quantity that has both magnitude (size) and direction.
✓ Scalar: A quantity with only magnitude (e.g., temperature, mass)
✓ Vector: A quantity with magnitude AND direction (e.g., velocity, force)
✓ Vectors are represented by arrows: length = magnitude, arrow points = direction
🧭 Compass Directions and Vectors
Cardinal Directions
The four main compass directions (cardinal points):
• North (N): 0° or 360°
• East (E): 90°
• South (S): 180°
• West (W): 270°
Angles are measured clockwise from North
Intercardinal Directions
• Northeast (NE): 45°
• Southeast (SE): 135°
• Southwest (SW): 225°
• Northwest (NW): 315°
📝 Example
A plane travels 200 km Northeast.
• Magnitude: 200 km
• Direction: 45° (Northeast)
📊 Component Form of a Vector
Definition
The component form expresses a vector in terms of its horizontal (x) and vertical (y) components.
v = ⟨x, y⟩ or v = ⟨a, b⟩
Finding Component Form from Two Points
Given initial point P(x₁, y₁) and terminal point Q(x₂, y₂):
v = ⟨x₂ - x₁, y₂ - y₁⟩
Unit Vector Notation (i, j)
Vectors can also be written using unit vectors:
v = ai + bj
• i = unit vector in x-direction = ⟨1, 0⟩
• j = unit vector in y-direction = ⟨0, 1⟩
📝 Example
Find the component form of vector from P(2, 3) to Q(7, -1).
v = ⟨7 - 2, -1 - 3⟩
v = ⟨5, -4⟩ or v = 5i - 4j
📏 Magnitude of a Vector
Definition
The magnitude (or length) of a vector is the distance from its initial point to its terminal point. It's always a non-negative number.
Formula
For vector v = ⟨a, b⟩ or v = ai + bj:
|v| = √(a² + b²)
This is based on the Pythagorean theorem
For 3D Vectors
For vector v = ⟨a, b, c⟩:
|v| = √(a² + b² + c²)
📝 Example
Find the magnitude of v = ⟨3, 4⟩.
|v| = √(3² + 4²)
|v| = √(9 + 16)
|v| = √25
|v| = 5
🔺 Direction Angle of a Vector
Definition
The direction angle θ is the angle the vector makes with the positive x-axis, measured counterclockwise.
Formula
For vector v = ⟨a, b⟩:
θ = tan⁻¹(b/a)
⚠️ Note: Adjust the angle based on the quadrant!
Quadrant Adjustments
• Quadrant I: θ = tan⁻¹(b/a)
• Quadrant II: θ = 180° + tan⁻¹(b/a)
• Quadrant III: θ = 180° + tan⁻¹(b/a)
• Quadrant IV: θ = 360° + tan⁻¹(b/a)
🎯 Component Form from Magnitude and Direction Angle
Formulas
Given magnitude |v| and direction angle θ:
a = |v| cos θ
b = |v| sin θ
Therefore: v = ⟨|v| cos θ, |v| sin θ⟩
📝 Example
Find component form of vector with magnitude 10 and direction angle 60°.
a = 10 cos 60° = 10 × 0.5 = 5
b = 10 sin 60° = 10 × (√3/2) ≈ 8.66
v = ⟨5, 8.66⟩ or v = 5i + 8.66j
➕ Vector Addition
Component Method
To add vectors, add their corresponding components:
u + v = ⟨a₁ + a₂, b₁ + b₂⟩
Where u = ⟨a₁, b₁⟩ and v = ⟨a₂, b₂⟩
Graphical Methods
1. Triangle Method (Tip-to-Tail):
• Place the tail of vector v at the tip of vector u
• Draw resultant from tail of u to tip of v
2. Parallelogram Method:
• Place both vectors at the same initial point
• Complete the parallelogram
• Diagonal from common point is the resultant
📝 Example
Add u = ⟨3, 5⟩ and v = ⟨-1, 2⟩.
u + v = ⟨3 + (-1), 5 + 2⟩
u + v = ⟨2, 7⟩
➖ Vector Subtraction
Component Method
To subtract vectors, subtract their corresponding components:
u - v = ⟨a₁ - a₂, b₁ - b₂⟩
Alternative Method
Subtracting v is the same as adding -v:
u - v = u + (-v)
To get -v, negate all components: -v = ⟨-a₂, -b₂⟩
📝 Example
Subtract v = ⟨2, 3⟩ from u = ⟨7, 1⟩.
u - v = ⟨7 - 2, 1 - 3⟩
u - v = ⟨5, -2⟩
✖️ Scalar Multiplication
Definition
Multiplying a vector by a scalar (number) multiplies each component by that scalar.
Formula
For scalar k and vector v = ⟨a, b⟩:
kv = ⟨ka, kb⟩
• If k > 0: Same direction, magnitude multiplied by k
• If k < 0: Opposite direction, magnitude multiplied by |k|
📝 Example
Find 3v where v = ⟨2, -1⟩.
3v = ⟨3×2, 3×(-1)⟩
3v = ⟨6, -3⟩
🎯 Resultant Vector
Definition
The resultant vector is the vector sum of two or more vectors. It represents the combined effect of all vectors.
Magnitude of Resultant (Parallelogram Law)
For two vectors P and Q at angle θ:
|R| = √(P² + Q² + 2PQ cos θ)
Special Cases
• Same direction (θ = 0°):
|R| = P + Q
• Opposite directions (θ = 180°):
|R| = |P - Q|
• Perpendicular (θ = 90°):
|R| = √(P² + Q²)
📋 Complete Vector Formula Summary
| Concept | Formula |
|---|---|
| Component Form | v = ⟨a, b⟩ or v = ai + bj |
| Component from Points | v = ⟨x₂ - x₁, y₂ - y₁⟩ |
| Magnitude | |v| = √(a² + b²) |
| Direction Angle | θ = tan⁻¹(b/a) |
| Component from Mag & Dir | v = ⟨|v| cos θ, |v| sin θ⟩ |
| Vector Addition | u + v = ⟨a₁ + a₂, b₁ + b₂⟩ |
| Vector Subtraction | u - v = ⟨a₁ - a₂, b₁ - b₂⟩ |
| Scalar Multiplication | kv = ⟨ka, kb⟩ |
| Resultant Magnitude | |R| = √(P² + Q² + 2PQ cos θ) |
💡 Quick Reference Tips
✅ Vectors have direction: Use arrows to represent them
✅ Component form: v = ⟨a, b⟩ gives horizontal and vertical parts
✅ Magnitude: Use Pythagorean theorem √(a² + b²)
✅ Adding vectors: Add corresponding components
✅ Unit vectors: i points right (x-axis), j points up (y-axis)
✅ Triangle method: Place vectors tip-to-tail for addition
✅ Compass angles: Measured clockwise from North (0°)
📚 Master vectors for success in Tenth Grade Math and Physics! 📚