🥋 Vectors – White to Black Belt Mastery
Intro Quote or Motto
“Direction without magnitude is like a punch without power. Both must be trained together.”
Key Concept (White Belt Level)
A vector is a quantity that has both magnitude and direction. Think of it like an arrow: the length shows its size, and the arrowhead shows where it points.
Core Principles
1. Magnitude is the length of the arrow.
2. Direction is the angle or bearing of the arrow.
3. Component form: A = (Aₓ, Aᵧ).
Common Mistakes and Pitfalls
Mixing up magnitude and direction. Use Pythagoras for magnitude and trig for direction.
Forgetting signs on components. Decide signs by the quadrant.
Adding magnitudes directly instead of components. Break vectors into x and y first.
Sensei’s Shortcuts
Draw a quick sketch to prevent sign errors.
Use SOH-CAH-TOA to break vectors into components.
Solve x and y separately, then recombine.
Worked Example – Step by Step (White Belt)
Problem
A vector has a magnitude of 10.0 m and points 37.0° above the positive x axis. Find its x and y components.
Solution
Break into components:
Aₓ = 10.0 × cos 37.0°
Aᵧ = 10.0 × sin 37.0°
Calculate (3 significant figures):
Aₓ = 10.0 × 0.799 = 7.99 m
Aᵧ = 10.0 × 0.602 = 6.02 m
Final Answer: A = (7.99 m, 6.02 m).
Practice Drill
A vector has a magnitude of 5.00 m and points 60.0° above the x axis. Find x and y components.
Find the magnitude and direction of B = (3.00, 4.00) m.
Add C = (4.00, 2.00) m and D = (-1.00, 5.00) m.
Answers
1. (2.50 m, 4.33 m)
2. 5.00 m at 53.1°
3. (3.00, 7.00) m
Yellow Belt Extension – Deeper Skills
Example: Adding two vectors given in magnitude and direction form
Problem
P = 8.00 m at 25.0°; Q = 6.00 m at 120°. Find R = P + Q .
Solution Outline
1. Convert each vector to components.
– Pₓ = 8.00 × cos 25.0°, Pᵧ = 8.00 × sin 25.0°
– Qₓ = 6.00 × cos 120°, Qᵧ = 6.00 × sin 120°
2. Add components: Rₓ = Pₓ + Qₓ, Rᵧ = Pᵧ + Qᵧ.
3. Magnitude: |R| = √(Rₓ² + Rᵧ²).
4. Direction: θ = tan⁻¹(Rᵧ / Rₓ), adjusted for quadrant.
Extra Practice
– Add U = 12.0 m at 40.0° to V = 10.0 m at 200°.
– Subtract S = (5.00, -3.00) m from T = (-2.00, 4.00) m.
Black Belt Mastery – Exam Strategy and Challenge
Challenge
A boat moves at 12.0 m/s east relative to the water. The water flows at 5.00 m/s north.
1. Find the boat’s velocity relative to the ground.
2. If the river is 300 m wide, how long to cross?
3. How far downstream will the boat land?
Sensei Strategy Notes
– Treat this as vector addition with perpendicular components.
– For part 2, use only the component perpendicular to the river bank.
– For part 3, use the crossing time with the parallel component.
Sensei’s Final Words
“Draw every vector before you calculate. The sketch is the stance; without it, your strike will miss.”
