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.
More to read
2.5 Equations of Lines and Planes in Space
A line in space is fixed by one point plus a direction vector, and a plane by one point plus a normal vector. From the point and direction come the vector, parametric and symmetric forms of a line, which all say the same thing in different notation. From the point and normal comes the plane equation ax + by + cz = d, where (a, b, c) is the normal. Converting between forms is mostly substitution and solving for the parameter. Distances and angles all 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. Two lines are parallel when their directions are multiples, intersecting when they share a point, and skew otherwise. The angle between planes is the angle between their normals.
2.2 Vectors in Three Dimensions
Every line and plane calculation rests on 3D vector arithmetic. Components add entry by entry, scalar multiples stretch or flip, and the length formula extends Pythagoras with a third square. Unit vectors along the axes give a clean basis for writing any displacement. Draw the box diagonal picture whenever a formula feels abstract. The dot product tests perpendicularity and builds projections, while the cross product builds a vector perpendicular to two given ones. Normals to planes are often found with a cross product of two in-plane directions. Master these two products and the geometry of space becomes routine algebra.
2.6 Quadric Surfaces
Planes are the simplest surfaces, and quadrics are the next step up: ellipsoids, paraboloids, cones, hyperboloids and their cousins. Each family has a standard equation whose signs and squares reveal its shape. Traces, which are slices parallel to the coordinate planes, turn a 3D mystery into familiar 2D curves. Sketch the traces first and the surface follows. Seeing quadrics trains the same skill that planes demand: reading geometry out of an equation. A missing variable means the shape extends straight along that axis. A minus sign flips a bowl into a saddle. This fluency with equations pays off when classifying intersections of lines and planes with curved surfaces.
Explore more
2 · Watch
Take it off screen
Where it sits
Learn first
This opens up
Nothing builds on it yet.
Where this leads
Jobs that lean on this skill. Follow one to see everything it is built on.
14 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.