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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.