Vector Calculator (Dot Product, Cross Product, Angle)

Both vectors are treated as 3D — leave the z components at 0 to work with plain 2D vectors, since a 2D vector is just a 3D vector with no z component. Magnitude is |V| = √(x² + y² + z²), the direct extension of the Pythagorean theorem to three dimensions.

The dot product, A · B = x₁x₂ + y₁y₂ + z₁z₂, is a single number related to how aligned the two vectors are — it's used here (via A · B = |A||B|cos θ) to solve for the angle between them: θ = cos⁻¹((A · B) ÷ (|A||B|)).

The cross product, A × B = (y₁z₂ − z₁y₂, z₁x₂ − x₁z₂, x₁y₂ − y₁x₂), is a vector perpendicular to both A and B, with magnitude equal to the area of the parallelogram they span. When both inputs are 2D (z = 0), only its z component is nonzero — that single number is the standard '2D cross product', commonly used to test which way one vector turns relative to another.