THE VECTOR LAB / GEOMETRY IN MOTION
Resolve a force. Find an angle. Drop a point onto a plane. Enter two vectors, then connect the algebra to an interactive, right-handed 3D view.
Drag to orbit · Focus the view, then scroll to zoom · Arrow keys rotate · + / − zoom
Arrows share a coordinate scale, except the display-scaled unit normal. The transparent plane patch illustrates an infinite plane through the origin, not a bounded surface.
| Quantity | Result |
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FROM EQUATION TO PICTURE
Add components to find a resultant vector. The dot product instead returns a scalar, useful for finding angles and resolving a vector along a direction.
A · B = AxBx + AyBy + AzBz
For a force in newtons and displacement in meters, their dot product is work in joules. This calculator does not assign physical units automatically.
Projection gives the part of A parallel to B. Subtract it from A to get the perpendicular remainder. B must be nonzero.
projB(A) = (A · B / |B|²) B
Choose “Resolve a vector”: A = (4,3,0), B = (2,0,0). The along component is (4,0,0) and the perpendicular component is (0,3,0).
A × B points perpendicular to both directions. Normalize it to define a plane normal. Project P onto that normal to get the signed distance to the plane through the origin.
n = (A × B) / |A × B|
d = P · n; Q = P − dn
The 3–4 triangle has area 6. Its plane is z = 0; P = (1,1,5) is 5 units away and its closest point is (1,1,0).
A zero vector has no direction. Projection onto zero and an angle involving zero are undefined. Parallel vectors do not span a unique plane; we also suppress plane calculations when the normalized cross-product magnitude is at most 10⁻¹⁰ because the plane is numerically unstable. Other valid results remain available.
The JSON contains all three input vectors, results, formulas, assumptions and numerical limits. CSV contains coordinate rows and scalar results. Undefined operations are null in JSON and explicitly marked undefined in CSV. Your inputs are not uploaded or stored by this tool.
Methods follow OpenStax Calculus Volume 3: the dot product and projections and the cross product. Uses double-precision arithmetic, not symbolic algebra. This is a learning and calculation aid, not engineering or safety certification.
Maintained by FastTool · Method reviewed October 14, 2026 · Report an issue