Find unknown parameters from vector equations confidently. Check intersections, parallel cases, magnitudes, and directional relationships. Study smarter using exports, formulas, examples, and step guidance.
| Dimension | A | B | C | D | Expected Result |
|---|---|---|---|---|---|
| 2D | ⟨1, 2⟩ | ⟨2, -1⟩ | ⟨5, 0⟩ | ⟨1, 1⟩ | λ = 1, μ = -2, point = ⟨3, 1⟩ |
| 3D | ⟨0, 1, 2⟩ | ⟨1, 2, 1⟩ | ⟨2, 5, 4⟩ | ⟨-1, 0, -1⟩ | Unique intersection with solved parameters |
| 3D | ⟨0, 0, 0⟩ | ⟨1, 1, 0⟩ | ⟨0, 1, 1⟩ | ⟨1, 1, 0⟩ | Parallel but distinct, no solution |
The solver uses the vector equation A + λB = C + μD. Rearranging gives λB - μD = C - A. This creates a linear system across coordinates.
For usable coordinate pairs, the calculator applies determinant-based solving. In 2D, two equations usually determine λ and μ directly. In 3D, the tool solves from a valid pair, then verifies the remaining coordinate for consistency.
It also computes supporting measures:
It solves A + λB = C + μD for unknown parameters. It also identifies unique intersections, parallel cases, inconsistent systems, magnitudes, dot products, angles, and direction relationships.
Yes. Choose 2D for x and y components only, or 3D for x, y, and z components. The solver automatically adjusts the equation checks.
A unique intersection means one pair of parameters makes both sides equal at the same point. The result includes λ, μ, and the shared coordinate point.
Parallel directions may produce infinitely many solutions or no solution. The calculator checks whether the offset between base vectors lies on the same direction line.
In 3D, two parameters may satisfy two coordinates but fail the third. That means the equation is inconsistent, often representing skew lines or mismatched vector conditions.
These values help interpret the geometry. Magnitudes show vector length, while the angle and dot product explain whether the directions are aligned, opposed, or orthogonal.
The residual check measures how closely both equation sides match after solving. A value near zero confirms a consistent computed solution.
Yes. After solving, use the CSV button for spreadsheet-friendly output or the PDF button for a printable summary of the solved result section.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.