Calculator
Example data table
| Case | Inputs | Direction d | k values | k·d |
|---|---|---|---|---|
| 2D (two points) | (1,2) → (5,8) | (4,6) | 1, 2 | (4,6), (8,12) |
| 2D (equation) | 3x − 2y + 7 = 0 | (−2, −3) | −1, 3 | (2,3), (−6, −9) |
| 3D (direction) | d = (1,2,3) | (1,2,3) | 0.5, 2 | (0.5,1,1.5), (2,4,6) |
Formula used
- Direction from two points: d = P2 − P1.
- 2D line equation: for ax + by + c = 0, a direction is (b, −a).
- Parallel vectors: any vector v = k·d is parallel for scalar k.
- Unit direction: u = d / |d|, where |d| = √(dx² + dy² [+ dz²]).
- Parallel test: 2D uses determinant dx·vy − dy·vx; 3D uses cross product magnitude |d×V|.
- Angle: θ = arccos( (d·V) / (|d||V|) ) in degrees.
How to use this calculator
- Select 2D or 3D, then choose a line definition method.
- Enter the required inputs (two points, a direction, an equation, or parametric coefficients).
- Provide one or more scale factors k to generate multiple parallel vectors.
- Optionally enter a point Q to build a parallel line through that point.
- Optionally enter a vector V to check parallelism and compute the angle.
- Click Submit to see results above the form, then download CSV or PDF.
Article
Choose the right line input
This tool accepts two points, a direction vector, a 2D equation, or a 3D parametric direction. Two points give d = P2 − P1. For ax + by + c = 0, the direction is (b, −a). In 3D, the direction is simply (a, b, c).
Turn a direction into many parallel vectors
A vector is parallel to the line when it is a scalar multiple: v = k·d. Enter several k values to generate a set of parallel vectors at once. Positive k keeps the same orientation, negative k flips it, and fractional k scales the length while preserving direction. When k = 0, the result is the zero vector, which has no direction.
Normalize for clean comparisons
The unit vector u = d/|d| removes scale and keeps only direction. This is useful when you want consistent comparisons between different inputs, or when k values are large or very small. Magnitude is computed as √(dx² + dy²) in 2D, and √(dx² + dy² + dz²) in 3D. A zero direction vector is rejected because it cannot define a line.
Build a parallel line through a point
If you provide a point Q, the calculator builds a parallel line using Q + t·d. This is a quick way to shift a line without changing its direction. In 2D, it also outputs a Cartesian form A x + B y + C = 0 for the shifted line, using a normal vector (−dy, dx).
Verify parallelism with a second vector
To test whether a vector V is parallel to the line’s direction, the tool checks a zero-area condition. In 2D it uses the determinant dx·vy − dy·vx, which equals zero when vectors are collinear. In 3D it uses the cross product magnitude |d×V|. Values near zero indicate parallel directions; use the angle to decide practical tolerance.
Angle output for tolerance checks
The computed angle θ comes from θ = arccos((d·V)/(|d||V|)). Even if two vectors are not exactly parallel due to measurement noise, a small angle confirms they are almost parallel. Use this for drawings, coordinate geometry exercises, and sanity checks. If θ is near 90°, the vectors are close to perpendicular.
Exportable results for reports
After submitting, you can export your direction vector, unit vector, scaled vectors, and optional parallel line form to CSV or PDF. This makes it easy to paste values into spreadsheets, lab notes, or homework solutions. The example table shows typical inputs and the corresponding k·d outputs.
FAQs
1) What does it mean for a vector to be parallel to a line?
A vector is parallel when it points in the same or opposite direction as the line’s direction vector. Mathematically, it must be a scalar multiple: v = k·d.
2) I have y = mx + b. How do I use it here?
Convert to ax + by + c = 0 by rewriting as mx − y + b = 0. Then enter a = m, b = −1, and c = b to get a parallel direction.
3) Why can’t the direction vector be zero?
The zero vector has no direction, so it cannot represent a line’s orientation. A valid line direction must have nonzero magnitude.
4) How should I choose k values?
Pick k values that match your needed lengths. Use k = 1 for the base direction, k = −1 for the opposite direction, and fractions like 0.5 for shorter parallel vectors.
5) What if the parallel test is “almost” zero?
Small nonzero values can come from rounding or measurement noise. Use the angle result as a tolerance check; a very small angle indicates practical parallelism.
6) Can this create a parallel line through a specific point?
Yes. Enter point Q and the calculator outputs a parallel line in parametric form Q + t·d. In 2D it also provides a Cartesian equation for easy graphing.