Intersections of Lines and Planes

The cardboard-and-straws phase of Where Planes Meet asked your group to build every way that lines and planes can share three-space, and the census surprised everyone. Two lines can cross, run parallel — or do something impossible on paper: miss each other entirely without being parallel, like two highways at different heights. Those are skew lines, and they are the first sign that three-space plays by richer rules. Two distinct planes never manage skewness: they are parallel, or they intersect — and when they intersect, they share a whole line, the crease your two cardboard sheets made. Three planes can meet in a point (the corner of the room), in a line (a book’s pages at the spine), in a plane, or nowhere at all.

The catalogue matters because every intersection question has two layers: what kind of meeting is possible, then where it happens. Algebra answers both — solutions of the combined equations are exactly the shared points, so the geometry is read off the algebra.

A line meets a plane

Substitute the line’s parametric form into the plane’s scalar equation and solve for the parameter. The line $\vec{r}=(1,1,0)+t(0,1,1)$ meets the plane $x+y+z=6$ where $1+(1+t)+t=6$, so $t=2$: the point $(1,3,2)$. One value of $t$, one crossing point. Had the variable vanished, the equation itself would have delivered the verdict: $0t=5$ means no solutions — the line is parallel to the plane and misses; $0t=0$ means every $t$ works — the line lies inside the plane. The algebra does not merely find the intersection; it diagnoses the configuration.

Three planes at once

Three scalar equations, three unknowns — elimination or substitution, exactly the systems machinery you already own, with a geometric reading attached to every outcome. A unique solution is a corner point; a one-parameter family is a spine line; a contradiction means at least two planes never meet.

Self-check: can you predict "no corner point" without

solving? Often, yes. If the three normals $\vec{a}$, $\vec{b}$, $\vec{c}$ satisfy $\vec{a}\cdot (\vec{b}\times \vec{c})=0$, the normals are coplanar and the three planes cannot pin down a single point — the configurations left are line, plane, or nothing. The test costs one cross product and one dot product; a solve that was doomed from the start costs a whole page. Check before you compute.

Distances belong to this family too — the distance from a point to a plane is measured along the normal, the perpendicular being the shortest path. It is the final move in The Flight Path: after the approach line meets the runway plane, you certify the clearance. Lines and Planes Practice closes with the full range — crossings, skew tests, corner points, and distances. Consolidating from the bottom, this page is the whole unit in one sentence: turn geometry into equations, solve, and translate the solution back into a picture.

Curriculum connection

C3.3

determine, through investigation using a variety of tools and strategies (e.g., modelling with cardboard sheets and drinking straws; sketching on isometric graph paper), different geometric configurations of combinations of up to three lines and/or planes in three-space (e.g., two skew lines, three parallel planes, two intersecting planes, an intersecting line and plane); organize the configurations based on whether they intersect and, if so, how they intersect (i.e., in a point, in a line, in a plane)

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C4.4

recognize a scalar equation for a plane in three-space to be an equation of the form $Ax+By+Cz+D=0$ whose solution points make up the plane, determine the intersection of three planes represented using scalar equations by solving a system of three linear equations in three unknowns algebraically (e.g., by using elimination or substitution), and make connections between the algebraic solution and the geometric configuration of the three planes Sample problem: Determine the equation of a plane $P_3$ that intersects the planes $P_1$, $x+y+z=1$, and $P_2$, $x-y+z=0$, in a single point. Determine the equation of a plane $P_4$ that intersects $P_1$ and $P_2$ in more than one point.

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C4.7

solve problems relating to lines and planes in three-space that are represented in a variety of ways (e.g., scalar, vector, parametric equations) and involving distances (e.g., between a point and a plane; between two skew lines) or intersections (e.g., of two lines, of a line and a plane), and interpret the result geometrically Sample problem: Determine the intersection of the perpendicular line drawn from the point $A(-5,3,7)$ to the plane $\vec{v}=(0,0,2)+t(-1,1,3)+s(2,0,-3)$, and determine the distance from point $A$ to the plane.

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