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Calculus IIIGrades 9–12University4 min read

Planes in Three Dimensions

A plane in three-dimensional space is determined by a point and a normal vector perpendicular to every direction in the plane.

Cheat sheet
A normal vector turns a flat surface in space into one linear equation.

Point-normal form

Let a plane pass through $P_0(x_0,y_0,z_0)$ with nonzero normal vector

$$ \vec n=\langle a,b,c\rangle. $$

For any plane point $P(x,y,z)$, the displacement $\overrightarrow{P_0P}$ lies in the plane and is perpendicular to $\vec n$. Therefore

$$ \vec n\cdot\overrightarrow{P_0P}=0, $$

or

$$ a(x-x_0)+b(y-y_0)+c(z-z_0)=0. $$

Scalar equation

Expanding point-normal form gives

$$ ax+by+cz=d. $$

The coefficients $a,b,c$ are components of a normal vector. Multiplying the entire equation by any nonzero scalar gives an equivalent equation for the same plane.

Worked example: point and normal

Plane through three points

Given noncollinear points $A,B,C$, form two in-plane vectors:

$$ \overrightarrow{AB},\qquad\overrightarrow{AC}. $$

Their cross product

$$ \vec n=\overrightarrow{AB}\times\overrightarrow{AC} $$

is normal to the plane. Then use any of the three points in point-normal form.

If the cross product is zero, the points are collinear and do not determine a unique plane.

Vector and parametric forms

If $\vec u$ and $\vec v$ are nonparallel directions in the plane, then

$$ \vec r=\vec r_0+s\vec u+t\vec v,qquad s,t\in\mathbb R. $$

Two independent parameters sweep across a surface. The normal is perpendicular to both direction vectors.

Relationships between planes

Planes are parallel when their normal vectors are scalar multiples. They are coincident if they are parallel and share a point or have proportional complete scalar equations.

Planes are perpendicular when their normal vectors are perpendicular:

$$ \vec n_1\cdot\vec n_2=0. $$

Two nonparallel planes intersect in a line.

Angle between planes

The acute angle between planes is the acute angle between their normals:

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

Use the absolute value because reversing a normal does not change the plane.

Intercepts and traces

Set two coordinates equal to zero to find an axis intercept when it exists. Set one coordinate to zero to find the plane's trace in a coordinate plane.

These features help visualize the surface, but missing intercepts can occur when the plane is parallel to an axis.

Point-to-plane distance

For plane $ax+by+cz=d$ and point $(x_0,y_0,z_0)$,

$$ D=\frac{|ax_0+by_0+cz_0-d|}{\sqrt{a^2+b^2+c^2}}. $$

The numerator measures signed displacement in the normal direction; division by the normal's magnitude normalizes it.

Applications

Planes represent walls, surfaces, linear constraints, and local approximations. Normal vectors appear in lighting, physics, computer graphics, and collision calculations.

State coordinate units and distinguish the infinite mathematical plane from a bounded physical surface.

Common mistakes

Using an in-plane direction as the normal. The normal must dot to zero with every plane direction.

Forgetting that proportional normals can describe parallel planes. Compare full equations to decide coincidence.

Using only two points to define a unique plane. Infinitely many planes contain one line.

Taking a cross product of collinear directions and accepting the zero vector as a normal. A normal must be nonzero.

Comparing plane angles without absolute value. Opposite normals represent the same orientation.

Quick self-check

  • Is the normal vector nonzero and perpendicular to known plane directions?
  • Does the equation contain the supplied point?
  • Are equivalent scalar multiples recognized?
  • If three points are used, are they noncollinear?
  • Do plane relationships follow from their normals?
  • Are intercepts, distance, and units interpreted correctly?
Check your understanding

Try it yourself

Hints are part of learning. Open one whenever it makes the next step feel possible.

1 practice question
Question 1Write a point-normal plane · Standard

Which plane passes through (2, −1, 4) with normal vector ⟨3, 2, −1⟩?

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