Equations of Planes

In Where Planes Meet, your group held a sheet of cardboard flat and stood a drinking straw perpendicular to it — and discovered that the straw is a better description of the sheet than the sheet is. Tilt the cardboard and the straw tilts with it; every direction lying in the sheet is perpendicular to the straw. That straw is the plane’s normal vector, and one normal plus one known point nails the plane completely.

The algebra is one dot product. A point $P$ lies on the plane exactly when the vector from the known point to $P$ is perpendicular to the normal $\vec{n}=(A,B,C)$ — a dot product of zero. Expand that condition and it tidies into the scalar equation:

$Ax+By+Cz+D=0$

The normal’s components sit in plain sight as the coefficients: one normal to $3x+5y-2z=6$ is $(3,5,-2)$, readable without any work. Every scalar multiple of a normal is another normal — the straw can be any length — which is why $6x+10y-4z=12$ is the same plane wearing doubled coefficients.

Building a plane from three points

Three points, as long as they refuse to line up, determine a plane — a camera tripod stands firm where a four-legged chair wobbles. The recipe: two vectors in the plane, then The Cross Product to manufacture the normal.

The plane through $(1,0,0)$, $(0,2,0)$, $(0,0,3)$

Two in-plane vectors, first point to the others: $\vec{u}=(-1,2,0)$ and $\vec{v}=(-1,0,3)$. The normal is $\vec{n}=\vec{u}\times \vec{v}=(6,3,2)$. With point $(1,0,0)$: $6x+3y+2z=6$. Audit before trusting — the second point gives $3(2)=6$, the third gives $2(3)=6$. All three check. An unaudited plane equation is a conjecture, not an answer.

Vector and parametric forms

Planes also take the point-plus-directions costume from Equations of Lines — but a plane is two-dimensional, so it needs two direction vectors and two parameters:

$\vec{r}={\vec{r}}_0+s\vec{u}+t\vec{v}$

Two independent sliders, a whole flat sheet of reachable points. To convert back to scalar form, cross the two direction vectors for the normal. One line either way — and both directions of travel are rehearsed in Lines and Planes Practice. What happens when planes and lines share the room — crossing, missing, colliding — is Intersections of Lines and Planes.

Curriculum connection

C3.2

determine, through investigation with technology (i.e., 3-D graphing software) and without technology, that the solution points $(x,y,z)$ in three-space of a single linear equation in three variables form a plane and that the solution points $(x,y,z)$ in three-space of a system of two linear equations in three variables form the line of intersection of two planes, if the planes are not coincident or parallel Sample problem: Use spatial reasoning to compare the shapes of the solutions in three-space with the shapes of the solutions in two-space for each of the linear equations $x=0$, $y=0$, and $y=x$. For each of the equations $z=5$, $y-z=3$, and $x+z=1$, describe the shape of the solution points $(x,y,z)$ in three-space. Verify the shapes of the solutions in three-space using technology.

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C4.3

recognize a normal to a plane geometrically (i.e., as a vector perpendicular to the plane) and algebraically [e.g., one normal to the plane $3x+5y-2z=6$ is $(3,5,-2)$], and determine, through investigation, some geometric properties of the plane (e.g., the direction of any normal to a plane is constant; all scalar multiples of a normal to a plane are also normals to that plane; three non-collinear points determine a plane; the resultant, or sum, of any two vectors in a plane also lies in the plane) Sample problem: How does the relationship $\vec{a}\cdot (\vec{b}\times \vec{c})=0$ help you determine whether three non-parallel planes intersect in a point, if $\vec{a}$, $\vec{b}$, and $\vec{c}$ represent normals to the three planes?

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C4.5

determine, using properties of a plane, the scalar, vector, and parametric equations of a plane Sample problem: Determine the scalar, vector, and parametric equations of the plane that passes through the points $(3,2,5)$, $(0,-2,2)$, and $(1,3,1)$.

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C4.6

determine the equation of a plane in its scalar, vector, or parametric form, given another of these forms Sample problem: Represent the plane $\vec{r}=(2,1,0)+s(1,-1,3)+t(2,0,-5)$, where $s$ and $t$ are real numbers, with a scalar equation.

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