Lines and Planes in Space
Write lines and planes in vector, parametric and Cartesian form, and answer distance and intersection questions with dot and cross products.
What a learner can do afterwards
- Convert a plane between its normal form and its parametric form
- Find the distance from a point to a line and to a plane
- Decide whether two lines meet, are parallel, or are skew, and find the angle where relevant
1 · Read
A line in space is fixed by one point plus a direction vector. From them come the vector, parametric, and symmetric forms, which all say the same thing. A plane is fixed by one point plus a normal vector, giving a(x minus x0) plus b(y minus y0) plus c(z minus z0) is 0, or ax plus by plus cz is d. Read the normal straight off the coefficients: 2x minus y plus 4z is 7 has normal (2, negative 1, 4). On the line r is (1, 0, 2) plus t(3, negative 1, 2), the x-coordinate at t is 2 is 7.
Distances flow from dot and cross products. The distance from a point to a plane divides the plane expression evaluated at the point by the length of the normal. The plane z is 2 is horizontal, so the point (0, 0, 5) sits a vertical gap of 3 above it. A plane with normal (1, 2, negative 1) through (3, 0, 1) satisfies 3 plus 0 minus 1 is 2, so k is 2.
Two lines are parallel when their directions are multiples, intersecting when they share a point, and skew otherwise. The line through the origin along x and the line through (0, 1, 0) along z never meet and are not parallel, so they are skew. The angle between two planes is the angle between their normals: x is 0 and y is 0 have perpendicular normals, so they meet at 90 degrees.
For skew lines, cross the direction vectors to get the common perpendicular, then project onto it for the distance. Convert between line and plane forms by substitution, solving for the parameter. Draw the box diagonal picture whenever a formula feels abstract.
Fix lines with a point and a direction, planes with a point and a normal, and let dot and cross products answer the rest.
2 · Watch
Take it off screen
Where it sits
Where this leads
Jobs that lean on this skill. Follow one to see everything it is built on.
8 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.