Lines and planes in space

A line is described as a motion: it starts at \(\mathbf r_0\) and moves in direction \(\mathbf d\) at the pace of the parameter \(t\). Separating coordinates gives the parametric equations \(x=x_0+td_1\), \(y=y_0+td_2\), \(z=z_0+td_3\). Two points \(P\) and \(Q\) suffice: the direction vector is \(Q-P\). Two lines in space can be parallel, meet, or be skew, that is, non-parallel with no common point, which is impossible in the plane.

A plane is described as a condition: its points are those for which \(\mathbf r-\mathbf r_0\) is perpendicular to \(\mathbf n\). With \(\mathbf n=\langle a,b,c\rangle\) it expands to \(ax+by+cz=d\); for example, the plane through \((1,2,3)\) with normal \(\langle2,-1,4\rangle\) is \(2x-y+4z=12\). The coefficients of the variables are thus the components of the normal.

When the data are three points or two directions, the normal comes from the cross product of two vectors in the plane. Two planes are parallel if their normals are proportional; otherwise they meet in a line whose direction is the cross product of the normals. A line and a plane are perpendicular when the direction vector is proportional to the normal.

To find where a line meets a plane, substitute its parametric equations into the plane equation and solve for \(t\). This pair of objects returns as soon as we differentiate: the tangent plane to a surface and the tangent line to a curve are their best linear approximation.