How this calculator works
All input forms are converted to points and direction vectors. Parallel directions give either coincident or disjoint lines. Otherwise a cross-product test checks whether the point displacement lies in the plane of both directions. Noncoplanar spatial lines are skew. Coplanar nonparallel lines have one intersection. Direction and displacement tests use a relative tolerance of 1e-12, so extremely close or nearly parallel lines may need exact methods.
Formula
For P + t U and Q + s V, let D = Q-P and C = U cross V. If C is nonzero and D dot C = 0, t = ((D cross V) dot C)/|C|^2.
Worked example
The equations x + y = 5 and x - y = 1 meet at (3, 2). In 3D, (t, 0, 0) and (0, s, 1) are skew because their z coordinates can never agree.
Advertisement
Frequently asked questions
Does this intersect segments or full lines?
It intersects the full infinite lines. A point outside either finite segment remains a valid line intersection.
Can vertical lines be entered?
Yes. Use standard or parametric form; slope form cannot express a vertical line.
What does coincident mean?
Both inputs describe the same infinite line, so there is no unique intersection point.