The Vector Gate

NODE 01

Two Worlds of Qty

5 kgvs

• Is knowing the number enough?
• Yes → Scalar (Mass, Time, Temp).
• No, direction matters → Vector (Force, Velocity).

NODE 02

Vector Language

• Arrow representation: Tail and Head.
• Length = Magnitude. Pointing = Direction.
• 5m East is fundamentally different from 5m West.

NODE 03

Vector Magnitude

|A| < |B|

• Length of the vector arrow.
• Positive magnitude or Zero vector.
• Magnitude notation: |A|.

NODE 04

Vector Direction

• Reference axis, Angle, CW/CCW.
• Compass vs Cartesian directions (+x, -y).

NODE 05

Vector Types

EQUAL

ANTI

• Zero, Unit, Equal, Negative vectors.
• Parallel, Anti-parallel, Collinear, Coplanar.

NODE 06

Vector Addition

• Geometric: Head-to-tail method.
• Resultant: from tail of 1st to head of 2nd.
• A + B = R.

NODE 07

Triangle Law

• If two sides of a triangle represent vectors...
• The third closing side represents the resultant.

NODE 08

Parallelogram Law

• Vectors with common starting point.
• Diagonal gives resultant.
• R = √(A² + B² + 2AB cos θ).

NODE 09

Special Cases

θ=0°

θ=180°

θ=90°

• Same direction: R = A + B.
• Opposite: R = |A - B|.
• Perpendicular: R = √(A² + B²).

NODE 10

Vector Subtraction

• A - B = A + (-B).
• Reverse the direction of B, then add.
• Essential for relative velocity.

NODE 11

Vector Resolution

• Breaking a single vector into components.
• Ax = A cos θ, Ay = A sin θ.
• Crucial tool for mechanics.

NODE 12

Rect Components

• A = Axi + Ayj + Azk.
• i, j, k represent unit vectors along axes.
• First doorway into 3D Physics.

NODE 13

Resultant from Comp

R = √(Rx² + Ry²)

• Rx = ΣAx, Ry = ΣAy.
• Direction: tan θ = Ry / Rx.
• The most reused tool in Physics.

NODE 14

Unit Vectors

i

j

k

• Magnitude = 1. Used to specify direction.
• |i| = |j| = |k| = 1.

NODE 15

Position Vector

• r = xi + yj + zk.
• Displacement: Δr = r2 - r1.
• Unifies kinematics into vector space.

NODE 16

Relative Position

• rAB = rA - rB.
• Position of A relative to B.
• Moving coordinate perspective.

NODE 17

Relative Velocity

• vAB = vA - vB.
• Same direction: relative speed = difference.
• Opposite direction: relative speed = sum.

NODE 18

River-Boat

• Boat velocity relative to water vs bank.
• Shortest path vs Minimum time.
• Resultant velocity and Drift.

NODE 19

Wind & Aircraft

✈

• Plane velocity relative to air + Wind.
• Ground velocity, Heading, Compensation angle.
• Vector addition as practical navigation.

NODE 20

2D Motion

• x-motion and y-motion are independent.
• Simultaneous motion.
• Solve as two 1D problems.

NODE 21

Projectile Preview

• Horizontal component (ux = u cos θ).
• Vertical component (uy = u sin θ).
• Gravity acts only downwards.

NODE 22

Trajectory

• Launch → Rise → Max Height → Fall → Land.
• Parabolic path.
• Horizontal velocity remains constant (idealized).

NODE 23

Projectile Equations

T = 2u sinθ / gH = u²sin²θ / 2gR = u²sin2θ / g

• Time of flight, Max height, Range.
• Assumes uniform gravity, no air resistance.

NODE 24

Special Cases

• Horizontal launch from height.
• Max range occurs at 45° (ideal level ground).
• Complementary angles yield same range.

NODE 25

Dot Product

• A · B = AB cosθ. (Scalar result).
• Extracts how much one vector lies along another.
• Crucial for Work & Energy (Realm 06).

NODE 26

Cross Product

• A × B = AB sinθ n̂. (Vector result).
• n̂ is perpendicular to the plane of A and B.
• Crucial for Torque and Magnetism.

NODE 27

Vector Diagrams

• Visual construction of additions, subtractions.
• Components, relative velocity.
• Physics is easier when drawn before calculated.

NODE 28

Misconceptions

• "Vectors are just numbers with arrows." → No.
• "Negative vector has negative magnitude." → No.
• Equal magnitude ≠ Equal vectors.

NODE 29

Solving Engine

• DRAW → CHOOSE AXES → RESOLVE.
• WRITE EQS → SOLVE x/y → RECOMBINE.
• A reusable habit for advanced mechanics.

NODE 30

The Vector Bridge

• Motion told us what happened.
• Vectors teach us where and which direction.
• Next: WHY does motion change? (Realm 04).