Vector Projection Calculator

How Vector Projection Calculator Works?

The Vector Projection Calculator compares two three-dimensional vectors, $A=(a_x,a_y,a_z)$ and $B=(b_x,b_y,b_z)$. It uses the dot product to determine how much of Vector A points in the direction of Vector B, then separates A into parallel and perpendicular components.

The magnitudes, or Euclidean lengths, of the vectors are calculated from their coordinates:

More precisely, the second magnitude is $\left|B\right|=\sqrt{b_x^2+b_y^2+b_z^2}$. The scalar projection of A onto B is the signed length of A along B:

The vector projection is obtained by multiplying Vector B by the scalar factor $\frac{A·B}{{\left|B\right|}^2}$:

Its x-, y-, and z-components are calculated by applying this factor to the corresponding components of B. The rejection is the part of A perpendicular to B:

The calculator also reports the projection magnitude, rejection magnitude, angle, and cosine similarity. The projection magnitude is the absolute value of the scalar projection, while the rejection magnitude is the length of R.

The angle between the vectors is the smaller angle between their directions, and cosine similarity is the normalized dot product:

Practical Use Cases

Students can use the calculator to check manual work in vector algebra, coordinate geometry, linear algebra, and physics exercises. Entering the six x-, y-, and z-components provides the dot product, magnitudes, angle, scalar projection, vector projection, and perpendicular rejection in one calculation.

Engineers can separate a three-dimensional force, velocity, displacement, or other vector into a component parallel to a reference direction and a component perpendicular to it. The vector projection identifies the directional contribution, while the rejection identifies the remaining transverse component.

Mathematicians can use the results to study orthogonal decomposition and verify relationships such as $A=proj{B}_A+R$, where the rejection is perpendicular to Vector B.

Computer graphics and robotics developers can project a position, movement direction, surface-related vector, or control vector onto a chosen axis in three-dimensional space. This helps identify motion along an axis separately from motion across it.

Physics and simulation users can compare directions through the angle and cosine similarity, then use the scalar and vector projections to analyze directional effects in a three-dimensional model.

Accuracy and Limitations

The calculator uses standard double-precision numerical arithmetic and reports rounded decimal results, so the final displayed values can differ slightly from exact hand calculations because of floating-point rounding. Each input component is limited to the range -1,000,000 through 1,000,000 with a 0.01 input step. Vector B must be nonzero because scalar and vector projection divide by $\left|B\right|$; a zero Vector B makes those results undefined. The angle and cosine similarity also require both vectors to be nonzero. Results are based only on the entered Cartesian x-, y-, and z-components and do not account for measurement uncertainty, coordinate-frame errors, or physical constraints.

Quick Examples