🥋 Vectors – White to Black Belt Mastery

Intro Quote

“Direction gives strength its purpose. Without it, even power is lost.”

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White Belt – Key Concept

Vectors have both magnitude and direction.
Scalars only have magnitude.

Examples:

• Velocity is a vector.
• Speed is a scalar.

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Core Principles

1. A vector can be written with an arrow symbol (A⃗).
2. Components break a vector into x- and y-parts.
3. Magnitude is found using Pythagoras.
4. Direction is the angle measured from the positive x-axis.
5. Vectors can be added graphically or algebraically.

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Key Equations (SI Units)

Magnitude
|A| = √(Aₓ² + Aᵧ²)

Direction
θ = tan⁻¹(Aᵧ / Aₓ)

Components from polar
Aₓ = |A| cos θ
Aᵧ = |A| sin θ

Resultant of two vectors
R⃗= A + B⃗ = (Aₓ + Bₓ, Aᵧ + Bᵧ)

Symbols:

• A⃗, B⃗ = vectors
• Aₓ, Aᵧ = components (m, m/s, N, etc.)
• θ = direction (degrees or radians)
• |A| = magnitude

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Common Mistakes and Pitfalls

• Forgetting the angle when describing a vector.
• Mixing up scalars and vectors.
• Wrong sign for components.
• Using tan⁻¹ without checking the quadrant.

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🥋 Sensei’s Shortcuts

• Always sketch the vector.
• Break into components before combining.
• Keep units consistent.
• Double-check the quadrant of the angle.

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Worked Example – Step by Step (White Belt)

Problem:
A⃗ has magnitude 12.0 m at 37° above the x-axis. Find Aₓ and Aᵧ.

Step 1. Formula
Aₓ = |A| cos θ
Aᵧ = |A| sin θ

Step 2. Substitution
Aₓ = 12.0 m × cos 37°
Aᵧ = 12.0 m × sin 37°

Step 3. Final Answer (3SF)
Aₓ = 9.58 m
Aᵧ = 7.22 m

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Practice Drill (with answers)

1. A⃗ has |A| = 15.0 m at 60°. Find Aₓ, Aᵧ.
   • Answer: Aₓ = 7.50 m, Aᵧ = 13.0 m
2. A⃗ = (8.00, 6.00) m. Find |A⃗| and θ.
   • Answer: |A| = 10.0 m, θ = 36.9°

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Yellow Belt Extension – Deeper Skills

Problem:
A⃗ = (10.0, 5.00) m
B⃗ = (−4.00, 12.0) m

Find R⃗, its magnitude, and its direction.

Solution Outline:

1. R⃗ = (10.0 − 4.00, 5.00 + 12.0) = (6.00, 17.0) m
2. |R| = √(6.00² + 17.0²) = 18.0 m
3. θ = tan⁻¹(17.0 / 6.00) = 70.3°

Final Answer (3SF):
R⃗ = (6.00, 17.0) m
|R| = 18.0 m
θ = 70.3°

Extra Practice:
C⃗ = (−7.00, −24.0) m → |C| = 25.0 m, θ = 253°

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Black Belt Mastery – Exam Strategy and Challenge

Problem:
A river flows east with velocity W⃗ = (2.50, 0) m/s.
A boat heads north at 3.00 m/s relative to the water.
Find the boat’s velocity relative to the ground.

Strategy Notes:

• Treat river and boat velocity as vectors.
• Add components.
• Find magnitude and direction.

Solution:
V⃗ = (2.50, 3.00) m/s
|V| = √(2.50² + 3.00²) = 3.91 m/s
θ = tan⁻¹(3.00 / 2.50) = 50.2° north of east

Final Answer (3SF):
V⃗ = (2.50, 3.00) m/s
|V| = 3.91 m/s
θ = 50.2° north of east

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Sensei’s Final Words

“A vector without direction is only half the story. Always train with both eyes open.”