Section 1 — What you will learn
✓ By the end you will understand
- Vectors as (x, y) components and as arrows (same thing).
- Dot product formula and why it measures “same direction”.
- Cosine similarity and why ML uses it for similarity.
- Projection (shadow) and what it means in ML.
- How changing arrows changes every number inside formulas.
Section 2 — Controls
Challenge mode gives a “cosine target” to hit.
Normalize ON is common in ML similarity search.
Target: 0.85 (higher = harder)
Auto Nudge slightly rotates your arrow to show cause→effect.
Section 3 — Live numbers
A = (ax, ay)
(0, 0)
B = (bx, by)
(0, 0)
|A| (magnitude)
0
|B| (magnitude)
0
Angle θ
0°
Cosine similarity
0.00
Section 4 — Formulas (LIVE)
∑
Watch how every term changes
Drag BLUE arrow
🧠 Kid Coach
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Tip: make the arrows point the same way (θ close to 0°).
Section 5 — Animated Vector World
A (YOU) — drag its tip
B (TARGET)
projection “shadow”
angle arc
Live explanation
Drag the BLUE arrow tip. Watch dot product & cosine update instantly.