Plane Intersections and Angles (HL only)

For a line \(\mathbf r=\mathbf a+t\mathbf d\) and plane \(\mathbf r\cdot\mathbf n=k\), substitute the line into the plane: \(\mathbf a\cdot\mathbf n+t(\mathbf d\cdot\mathbf n)=k\).

The line–plane angle \(\alpha\) satisfies \(\sin\alpha=\dfrac{|\mathbf d\cdot\mathbf n|}{|\mathbf d||\mathbf n|}\). The smaller angle \(\theta\) between two planes satisfies \(\cos\theta=\dfrac{|\mathbf n_1\cdot\mathbf n_2|}{|\mathbf n_1||\mathbf n_2|}\). Both lie in \(\left[0,\dfrac\pi2\right]\).

Two nonparallel planes meet in a line with direction \(\mathbf n_1\times\mathbf n_2\). Find one common point by solving the equations, then write the line through that point. Parallel normals require a check for coincident versus distinct parallel planes.

Three planes

Solve all three Cartesian equations together. One unique solution represents one common point; one free parameter represents a common line; two free parameters represent the same plane; inconsistency means there is no common point. Pairwise intersections do not guarantee a point common to all three.

Example

The planes \(x+y+z=3\), \(x-y=1\), \(z=0\) meet at \((2,1,0)\). Replacing the third equation by \(2x+2y+2z=6\) gives a common line; replacing it by \(2x+2y+2z=7\) gives no common point.

A third plane and an intersection line

When two non-parallel planes meet in \(\mathbf r=\mathbf p+t\mathbf d\), substitute that entire line into a third plane \(\mathbf n\cdot\mathbf r=c\). The common points satisfy \((\mathbf n\cdot\mathbf d)t=c-\mathbf n\cdot\mathbf p\).

If \(\mathbf n\cdot\mathbf d\ne0\), there is one parameter value and one common point. If \(\mathbf n\cdot\mathbf d=0\) and \(c-\mathbf n\cdot\mathbf p=0\), the whole line lies in the third plane. If the first expression is zero but the second is not, no point of the line lies in that plane: the three-plane system is inconsistent.

Check the zero-coefficient case before dividing. A redundant equation means dependence, not inconsistency; an impossible equation means inconsistency. Finally substitute a proposed point or line into all original equations.