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Planes
Planes and vector geometry
CatalogueDPMathematicsMathematics: Analysis and Approaches HLGeometry and trigonometry

Planes and vector geometry

Represent planes with normals and direction vectors, then solve intersections and angle problems.

  • AA HL
  • Planes
  • Papers 1 and 2
  • GDC optional
  • About 25 min

By the end of this lesson

  • move between vector and Cartesian plane forms
  • find line–plane intersections
  • classify plane relationships and calculate angles

Why this matters. A plane can be described by points and directions or by a normal. Choosing the right representation turns spatial geometry into manageable algebra.

Two views of a plane

A plane through point $\mathbf a$ with independent directions $\mathbf u$ and $\mathbf v$ has vector form

$$\mathbf r=\mathbf a+\lambda\mathbf u+\mu\mathbf v.$$

If $\mathbf n=\mathbf u\times\mathbf v$ is normal to the plane, then $\mathbf n\cdot(\mathbf r-\mathbf a)=0$, giving $Ax+By+Cz=D$.

Plane and vector geometryA slanted plane contains direction vectors u and v. A normal vector n rises perpendicular to the plane, while a line crosses the plane at point P.uvnP
The normal is perpendicular to every direction in the plane. A line–plane intersection satisfies both equations.

Checkpoint

For $2x-y+3z=7$, which vector is normal to the plane?
Show explanation

The Cartesian coefficients form a normal vector.

Example 1

Build a Cartesian equation

Point and two directions

A plane passes through $A(1,0,2)$ and contains directions $\mathbf u=(1,1,0)$ and $\mathbf v=(0,2,1)$.

$$\mathbf n=\mathbf u\times\mathbf v=(1,-1,2).$$

Using $\mathbf n\cdot(\mathbf r-\mathbf a)=0$ gives

$$(x-1)-y+2(z-2)=0\Rightarrow x-y+2z=5.$$

Intersect a line with a plane

Substitute the parametric coordinates of the line into the plane equation. One parameter value gives one intersection point.

Checkpoint

A line has direction $\mathbf d$ and a plane has normal $\mathbf n$. When is the line parallel to the plane?
Show explanation

A direction lying in or parallel to the plane is perpendicular to its normal.

Example 2

Find an intersection

Substitute one parameter

The line $\mathbf r=(1,2,0)+t(2,-1,3)$ meets $x+2y-z=4$.

Substitute $x=1+2t$, $y=2-t$, $z=3t$:

$$(1+2t)+2(2-t)-3t=4\Rightarrow t=\frac13.$$
$$P=\left(\frac53,\frac53,1\right).$$

Substitution into the plane equation checks the result.

Use normals to compare planes

Parallel normals give parallel or coincident planes. Non-parallel normals give an intersection line. The acute angle $\theta$ between planes satisfies

$$\cos\theta=\frac{|\mathbf n_1\cdot\mathbf n_2|}{|\mathbf n_1||\mathbf n_2|}.$$

Checkpoint

Planes with normals $(1,2,-1)$ and $(2,4,-2)$ are necessarily…
Show explanation

The normals are parallel. Compare constants to decide whether the planes coincide.

Example 3

Find an angle

Angle between two planes

Find the acute angle between $x+y=2$ and $x-y+z=0$.

Use normals $\mathbf n_1=(1,1,0)$ and $\mathbf n_2=(1,-1,1)$.

$$\mathbf n_1\cdot\mathbf n_2=1-1+0=0.$$

The normals are perpendicular, so the planes meet at $90^\circ$.

This conclusion needs no decimal approximation.

In the exam

Quick check

Five questions to check the main decisions from this lesson.

Your score is not saved.

  1. A normal to $3x+z=5$ is…
    Show explanation

    Read the coefficients of $x,y,z$.

  2. Two independent vectors in a plane produce a normal through…
    Show explanation

    The vector product is perpendicular to both directions.

  3. A line–plane intersection is found by…
    Show explanation

    The intersection must satisfy both representations.

  4. If $\mathbf n_1\cdot\mathbf n_2=0$, the planes are…
    Show explanation

    Their normals, and therefore the planes, meet at a right angle.

  5. Parallel normals and unequal compatible constants describe…
    Show explanation

    Proportional left sides with inconsistent constants cannot coincide.

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