Vector equations of lines in 3D – intersecting, parallel and skew lines
MathematicsVectors & Coordinate GeometryAges 17–18
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Sign in to playA line r = a + λb in a rotatable 3D coordinate frame (2D projection or 3D view): drag the point P(λ) along the line and read its parametric and Cartesian equations. Two lines are classified as intersecting, parallel, coincident or skew with step-by-step working (equate components, solve two equations, check the third), the angle between them, the point of intersection and the shortest distance. Further modes find the foot of the perpendicular from a point, where a line meets a plane and at what angle, and use r = a + tv for constant-velocity motion to find when two objects are closest.
Lesson: Vector equation of a line in space; intersecting, parallel and skew lines; angles and distances
What it shows
In three dimensions a line is written r = a + λb, where a is the position vector of a point on it and b is a direction vector. Unlike in a plane, two lines in space need not meet even if they are not parallel: such lines are skew. Equating the components gives three equations in λ and μ; solving two and checking the third decides whether the lines intersect. The scalar product gives the angle between lines, and the vector product gives the shortest distance between skew lines. With r = a + tv the same equation describes constant-velocity motion.
How to use
Choose a Mode and type the coordinates of a, b, c and d, or pick an Example. Drag the background to rotate, and drag P(λ) or Q(μ) along their lines or use the λ and μ sliders. The working below the picture updates step by step. Switch between 2D and 3D; Initial view resets the angle. In the motion mode, press Play to move the objects and watch the distance graph.
Parameters you can change
- Mode One line r = a + λb, Two lines: intersecting, parallel or skew, Distance from a point to a line, A line and a plane, Constant-velocity motion r = a + tv
- a (point A, start of object A) – x-coordinate -10–10
- a – y-coordinate -10–10
- a – z-coordinate -10–10
- b (direction vector, velocity of object A) – x-component -10–10
- b – y-component -10–10
- b – z-component -10–10
- c (point C on L₂, point Q, point on the plane, start of object B) – x-coordinate -10–10
- c – y-coordinate -10–10
- c – z-coordinate -10–10
- d (direction of L₂, normal of the plane, velocity of object B) – x-component -10–10
- d – y-component -10–10
- d – z-component -10–10
- Starting value of λ (time t in the motion mode) -10–10
- Picture 3D (three.js), 2D (rotatable projection)
Questions to explore
- Two lines in space are not parallel and do not meet. Why is that impossible in a plane but possible here?
- Drag P and Q to make |PQ| as small as possible. How is the segment PQ then related to both lines?
- If the paths of two moving objects cross, must the objects collide? Use the examples to decide.
