Visualize addition, subtraction, dot & cross products in 2D
Addition/Subtraction: Combine vectors tip-to-tail. Scalar multiplication: Scale magnitude, flip for negatives. Dot product: u · v = |u||v|cos θ — measures alignment (positive = same direction, zero = perpendicular, negative = opposite). Cross product in 2D gives a scalar (the z-component): |u × v| = |u||v|sin θ — measures the area of the parallelogram they span. Projection: proj_v(u) is the shadow of u onto v.