Lines in Three-Dimensional Space
Main Concept
In 2-dimensional Euclidean space, if two lines are not parallel, they must intersect at some point. However, this fact does not hold true in three-dimensional space and so we need a way to describe these non-parallel, non-intersecting lines, known as skew lines.
A pair of lines can fall into one of three categories when discussing three-dimensional space:
Intersecting: The two lines are coplanar (meaning that they lie on the same plane) and intersect at a single point.
Parallel: The two lines are coplanar but never intersect because they travel through different points, while their direction vectors are scalar multiples of one another.
Skew: The two lines are not coplanar but instead lie on parallel planes. This means they will never cross and do not have similar direction vectors. Because they lie on parallel planes, which share the same normal vector, the distance between the skew lines at their closest point can be calculated as the scalar projection of a vector pointing from one line to another onto the common normal vector.
Use the sliders below to define Line 1 and Line 2 by providing a point and direction vector from which they can be drawn. Then, mark the checkboxes below: "Show Points and Vectors", "Show Plane(s)" and "Show Normal Vector of Plane" to compare the points and vectors that make up these lines, the planes they line on, and the normal vectors of the planes, respectively. Try rotating the plot to get a better view of the lines and planes in three-dimensional space.
Line 1
Line 2
Point A
Point B
xA =
xB=
yA=
yB=
zA=
zB=
Direction Vector u
Direction Vector v
xu =
xv =
yu =
yv =
zu =
zv =
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