Vector Projection Calculator

What does this vector projection calculator compute?

proj_v u = ((u · v) / |v|²) v.

The published formula or method is proj_v u = ((u · v) / |v|²) v. The following references support the stated mathematics or method:

• Gilbert Strang, Calculus — Chapter 11: Vectors and Matrices (MIT OpenCourseWare): Vector components and magnitudes, dot product and angle/projection relations, and cross product/determinant orientation and area. Projecting a point displacement onto the unit normal to Ax+By+C=0 gives distance |Ax₀+By₀+C|/sqrt(A²+B²); this planar expression is derived from the projection formula. Mathematical reference only; not implementation validation or an endorsement of CalculatorX.

Which inputs and limits should you check?

Follow the input domain, units, branch or solve direction, and rounding policy in the specification. Preserve exact outputs and any additional candidates or warnings.

The worked example uses u = (2, 2, 0), v = (1, 0, 0) with ((2·1)/1) (1, 0, 0). Compare it with the Example calculations and the complete result for the same inputs.

How can you verify and cite this result?

CalculatorX is responsible for this page. The mathematical references above explain the model; implementation checks and their scope are documented separately in the math.vector v1.8.0 Evidence and capability contract. Read the displayed verification and production state together with the specification's numerical limits.

Use the clean calculator URL to cite the tool. A shared input link identifies a particular calculation. Content updates, numerical verification, and domain reviews have separate dates and scopes.

Numerical range, precision and figures

Inputs accept finite decimals, including scientific notation. Required x/y components cannot be empty. z defaults to zero only when absent from a request; an explicitly empty z is invalid. Nonzero inputs and every reported numeric field must be normal binary64 (magnitude at least 2⁻¹⁰²²). Overflow or a nonzero result below this range returns NUMERIC_RANGE; rescale the vectors. Decimal text which rounds to zero returns NUMERIC_UNDERFLOW instead of silently becoming zero.

Addition, subtraction, dot, cross and scalar triple products use exact dyadic intermediates followed by binary64 rounding. Projection uses exact intermediate ratios. Magnitude uses scaled hypot; normalization and the atan2 cross/dot angle are approximate. Results retain binary64 precision. Reference comparisons use relative tolerance 2×10⁻¹² (or 8 ULP), with absolute 2×10⁻¹⁰ degrees for ordinary angles and 2×10⁻¹³ for cosine. Cancellation can make relative component error sensitive; directions are undefined at zero and sensitive near zero. This is bounded vector arithmetic, not a general arbitrary-precision or symbolic system.

Units must be consistent across components: magnitude, sum, difference and projection retain component units; dot/cross products square them; the scalar triple product cubes them. Normalized components and projection scale are dimensionless. Angles include degrees, radians and cosine. Zero u can be projected onto nonzero v; normalizing zero, finding its direction angle, or projecting onto zero v is invalid.

Arrow figures show only the xy plane when every relevant z is exactly zero; cross/triple and 3D results use complete text and worked results. Arrows share a linear scale. Nonzero arrows smaller than a pixel receive a “below plot resolution” label instead of a false zero marker.

Interactive calculations run locally. Shared links contain the entered components and may be server-rendered. Exported JSON/CSV preserves the entered inputs, full numeric result and calculation version. Treat shared links and downloaded files as copies of your data.