2D and 3D Vector Calculator

Vectors in the plane and in space, with exact results and a drawing.

The calculator works with 2D and 3D vectors typed as components or as magnitude and angle. From A and B it computes the sum, the difference, k·A, the magnitude, the unit vector, the direction, the dot product, the angle between them, the cross product (with the parallelogram and triangle areas), and the projection and perpendicular component of A on B. In 3D, with a third vector C, it adds the triple product and the volume of the parallelepiped and the tetrahedron. When the components are numbers or fractions the result is exact and magnitudes come out with a simplified radical: for A = (3, 4) and B = (-1, 2), |B| = √5, A · B = 5 and cos θ = √5/5, or 63.4349°. The Resultant of forces mode adds several forces given by magnitude and angle, shows the resultant and the equilibrant force, and draws the head-to-tail polygon. In 3D the drawing is an orthographic projection with adjustable rotation and camera height.

Type the vectors, pick the drawing and check every step.

1. In Vector operations, choose 2D or 3D and type the components of A and B (or switch to magnitude and angle). The calculator opens with A = (3, 4) and B = (-1, 2).
2. Read the cards grouped as sum and magnitude, products and projection, open Step by step to see the formula and substitution, and switch the drawing between parallelogram, head to tail, difference, projection and cross product.
3. In Resultant of forces, enter the magnitude and angle of each force, pick a unit and read the resultant, the equilibrant force and the table of Fx and Fy components.

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Sources and references for this tool

These references help contextualize formulas, standards, APIs and limitations used on this page. They do not replace professional validation when a result has legal, financial, medical or operational impact.

• The Dot Product (Calculus Volume 3, 2.3)OpenStax, Dot product, angle between vectors, orthogonality and projection.
• The Cross Product (Calculus Volume 3, 2.4)OpenStax, Cross product, parallelogram area and the triple scalar product.

Frequently asked questions

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