Plane

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Distance between parallel planes ( $∣ (a_{1} - a_{2}) \cdot n ∣$ )

Planes with common unit normal

n

∣ (a_{1} - a_{2}) \cdot n ∣

. Every joining displacement has this normal component, giving a lower bound by its absolute value; a displacement parallel to the normal attains the bound.

Line-plane intersection criterion

A line

r = b + s m

intersects a plane

(r - a) \cdot n = 0

exactly when the scalar equation

s (m \cdot n) = (a - b) \cdot n

can be solved. If

m \cdot n \neq = 0

, there is one point. If that product vanishes, the line is either contained in the plane or disjoint from it. Direction and normal vectors must be nonzero.

Equation of a plane through three points ( $(r - a) \cdot [(b - a) \times (c - a)] = 0$ )

For three non-collinear points with position vectors

a, b, c

, a nonzero normal vector is

N = (b - a) \times (c - a) .

The plane is the set of

r

satisfying

(r - a) \cdot N = 0

. Expanding the cross product and using the scalar triple product gives

r \cdot (a \times b + b \times c + c \times a) = a \cdot (b \times c) .

Non-collinearity is essential: otherwise

N = 0

and the equation fails to select a unique plane.

Euclidean plane ( $R^{2}$ )

The Euclidean plane is

R^{2}

with its standard Euclidean norm and the resulting distance, angles, and Riemannian metric.

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