Vector Math & 3D Dot/Cross Product Calculator

The dot product of two vectors a and b yields a scalar value. Algebraically, it's the sum of component products: a · b = a_x b_x + a_y b_y + a_z b_z Geometrically, it connects vector magnitudes with the cosine of the enclosed angle θ (0 ≤ θ ≤ π): a · b = ‖a‖ ‖b‖ cosθ

This calculator uses that relationship to solve for the angle between the two input vectors directly: θ = arccos( (a · b) / (‖a‖ ‖b‖) )

Two non-zero vectors are orthogonal (perpendicular, θ = 90°) if and only if a · b = 0. In physics, mechanical work W done by a force F along a displacement d is the dot product W = F · d.

This calculator does not compute vector projections directly, but you can derive one by hand from its dot product and magnitude outputs.

The scalar projection of a onto b is the signed length of a's shadow cast onto b: comp_b a = (a · b) / ‖b‖

The vector projection is the vector parallel to b with that scalar length: proj_b a = ( (a · b) / ‖b‖² ) b

The vector rejection, the component of a perpendicular to b, is a - proj_b a. This decomposition is used in the Gram-Schmidt process for building orthogonal bases, and in physics to split forces into parallel and perpendicular components on an inclined plane.

The cross product of two 3D vectors a and b yields a new vector c = a × b that is perpendicular to both, computed via a determinant expansion: a × b = |i, j, k; a_x, a_y, a_z; b_x, b_y, b_z| = (a_y b_z - a_z b_y)i - (a_x b_z - a_z b_x)j + (a_x b_y - a_y b_x)k

Geometrically, the magnitude of the cross product equals the area of the parallelogram formed by a and b: ‖a × b‖ = ‖a‖ ‖b‖ sinθ

Its direction follows the right-hand rule: curl your right-hand fingers from a to b, and your thumb points along a × b. The cross product is anti-commutative: b × a = -(a × b). This calculator always evaluates the cross product using all three components. If you're in 2D mode, the z-components default to 0.

This calculator works with two vectors at a time, so triple products (which need a third vector c) aren't computed directly, but you can combine two calculator runs to get there by hand.

The scalar triple product a · (b × c) is a determinant whose absolute value equals the volume of the parallelepiped formed by the three vectors; if it equals zero, the vectors are coplanar. The vector triple product a × (b × c) is evaluated using the "BAC-CAB" identity: a × (b × c) = b(a · c) - c(a · b)

Choose 2D or 3D mode and an operation (add, subtract, dot, or cross), then enter the components of Vector A and Vector B. The calculator shows the result, both vector magnitudes, and (for the dot product) the angle between the vectors.

Using the calculator's default 2D vectors:

A + B = (3+1, 4+2) = (4, 6)

‖A‖ = √(3²+4²) = 5, ‖B‖ = √(1²+2²) ≈ 2.236

A · B = 3(1) + 4(2) = 11, θ = arccos(11 / (5 × 2.236)) ≈ 10.3°

With z=0 for both vectors, the cross product is A × B = (0, 0, 2), matching what the calculator returns for its default inputs.

For linear systems and determinants beyond two vectors, use the matrix solver. To find angle coordinates on a unit circle rather than between two vectors, see the unit circle calculator.

The cross product describes several rotational and electromagnetic quantities. Torque τ = r × F is the rotational force created about a pivot by a force F applied at position r. Angular momentum L = r × p describes a particle's rotational momentum. The Lorentz magnetic force F_B = q(v × B) acts on a moving charge q inside a magnetic field B, always perpendicular to both velocity and field. This is why charged particles follow helical paths in magnetic fields.

You can use this calculator's cross-product mode to work any of these three formulas by hand, entering the two relevant vectors as A and B.