2D Vectors

A 2D vector represents position, velocity, or direction. (x, y) tuple.

Common operations:

• Length: sqrt(x² + y²) (Pythagorean).
• Length squared: x² + y² (cheaper; use for comparisons).
• Normalize: v / length(v) → unit vector.
• Dot product: v · w = vx*wx + vy*wy. Measures alignment. 0 = perpendicular.
• Cross product (2D scalar): v × w = vx*wy - vy*wx. Sign tells side.

Geometric meanings:

• dot(a, b) = |a| * |b| * cos(θ): angle between.
• dot(a, b) > 0: same direction.
• dot(a, b) = 0: perpendicular.
• dot(a, b) < 0: opposite directions.
• cross(a, b) > 0: b is counterclockwise from a.

Projection: proj_b(a) = (a · b̂) * b̂ (a's component along b).

Reflection: r = v - 2 * (v · n̂) * n̂ (v reflected off surface with normal n).

Used in: bouncing balls.

Performance:

• Avoid sqrt when possible (use squared lengths).
• Inline simple ops.
• For large simulations, use SIMD or GPU.

NumPy works for batched vectors:

For 1000+ bodies: NumPy vectorization is essential.