A line in 3D is parametrized as r(t) = r₀ + td (position plus scalar multiple of direction vector). A plane with normal vector n = ⟨a, b, c⟩ passing through (x₀, y₀, z₀) has equation a(x−x₀) + b(y−y₀) + c(z−z₀) = 0.
In 2D, a line is fully described by a slope and a point — but in 3D, there is no single "slope." Instead, the natural description of a line uses the idea from your study of vectors: start at a known point and walk in a fixed direction. The parametric equation of a line is r(t) = r₀ + td, where r₀ is the position vector of a point on the line and d is the direction vector. As t ranges over all real numbers, r(t) traces out the entire line. The parameter t plays the role of a signed distance along the line. In component form: x = x₀ + at, y = y₀ + bt, z = z₀ + ct, where ⟨a, b, c⟩ = d. Two parallel lines have proportional direction vectors; two intersecting lines share a point for some pair of t values.
A plane in 3D is characterized not by a direction of travel but by a normal vector — a vector perpendicular to every vector lying in the plane. This is where the cross product becomes essential: if you know two vectors u and v that lie in the plane, then n = u × v is normal to both and hence to the plane. Given a normal vector n = ⟨a, b, c⟩ and a point (x₀, y₀, z₀) in the plane, any other point (x, y, z) in the plane satisfies: the vector from (x₀, y₀, z₀) to (x, y, z) must be perpendicular to n. Writing this dot product condition gives the point-normal form: a(x−x₀) + b(y−y₀) + c(z−z₀) = 0, which expands to ax + by + cz = d for some constant d. The coefficients of x, y, z in the plane equation are precisely the components of the normal vector.
The dot product plays a structural role throughout. The distance from a point P to a plane is computed by projecting the vector from any plane point to P onto the unit normal — it's |(P − P₀) · n̂|. Two planes are parallel if their normals are parallel (proportional), perpendicular if their normals are orthogonal (dot product zero), and intersect in a line otherwise. The line of intersection of two planes is found by solving the two plane equations simultaneously, and its direction vector is the cross product of the two normals — perpendicular to both.
These representations feed directly into the tangent plane problems ahead. When you compute the tangent plane to a surface z = f(x, y) at a point, you are finding the plane that best approximates the surface — and its normal vector will come from the partial derivatives. The parametric line equation appears in optimization (moving along a line to find minima) and in computing intersections needed for integration. Mastering the point-normal form and the parametric line form, and the geometric role of dot and cross products in deriving them, provides the foundation for nearly every 3D calculation in multivariable calculus.
No topics depend on this one yet.