Vector Component Resolver — 2D & 3D

Resolving Vectors: Components, Projection, and Rejection

Vectors show both size and direction, which makes them essential for everything from physics homework to robotics and game development. This calculator helps you break a vector into clear, usable parts: its components, magnitude, unit vector, and (when you add a reference vector) its projection and rejection. If you are working in 2D or 3D coordinates, the tool gives an instant, readable breakdown so you can focus on understanding the geometry rather than doing repetitive calculations by hand.

In Cartesian coordinates, a 2D vector is written as a = ⟨aₓ, aᵧ⟩ and a 3D vector as a = ⟨aₓ, aᵧ, a_z⟩. The magnitude (or length) tells you how long the vector is: |a| = √(aₓ² + aᵧ²) in 2D and |a| = √(aₓ² + aᵧ² + a_z²) in 3D. The unit vector â = a / |a| keeps the direction while normalizing the length to 1, which is useful in navigation, physics, and computer graphics.

Often you need a vector’s component along another direction. If you provide a reference vector b, the calculator splits a into a parallel part and a perpendicular part: proj_b(a) = (a·b / |b|²) b and rej_b(a) = a − proj_b(a). This is the same idea used to resolve forces on a ramp, align a velocity with a heading, or separate a signal into a preferred basis.

You can enter vectors either by components or by magnitude and angles. In 2D, an angle θ from the positive x-axis gives a = |a|⟨cos θ, sin θ⟩. In 3D, the calculator uses azimuth φ in the x–y plane and elevation θ above it: a = |a|⟨cos θ cos φ, cos θ sin φ, sin θ⟩.

How to use the vector component resolver

1. Choose 2D or 3D mode.
2. Enter vector a by components or by magnitude and angles.
3. Optionally enter a reference vector b to compute projection and rejection.
4. Click calculate to see components, magnitude, unit vector, and the angle between vectors.
5. Use the preview (2D) or LaTeX/Share buttons to reuse results in notes or reports.

Real-world examples include decomposing wind velocity into headwind and crosswind, resolving forces in statics problems, computing a robot’s motion along a desired path, or measuring how closely two directions align using the angle from the dot product. All calculations run locally in your browser for speed and privacy, and the output is formatted for quick copying into homework, documentation, or design notes.