Angle Between Two Planes⚠ unverified

Understanding the Parameters

• A_1 — n1 x.
• B_1 — n1 y.
• C_1 — n1 z.
• A_2 — n2 x.
• B_2 — n2 y.
• C_2 — n2 z.
• Output θ (degree) — Angle.

How to Calculate

1. Enter n1 x as A1 (default 1 dimensionless). Use the unit menu when you need a different unit.
2. Enter n1 y as B1 (default 0 dimensionless). Use the unit menu when you need a different unit.
3. Enter n1 z as C1 (default 0 dimensionless). Use the unit menu when you need a different unit.
4. Enter n2 x as A2 (default 0 dimensionless). Use the unit menu when you need a different unit.
5. Enter n2 y as B2 (default 1 dimensionless). Use the unit menu when you need a different unit.
6. Enter n2 z as C2 (default 0 dimensionless). Use the unit menu when you need a different unit.
7. Click Calculate. The card evaluates $\cos\theta=|\mathbf{n}_1\cdot\mathbf{n}_2|/(|n_1||n_2|)$ and shows the result in the declared output unit; Result in (when present) converts that number.

The Science: A Rigorous Derivation (Approaching a Proof)

Analytic geometry replaces figures with equations. A point is a tuple of real numbers; the Euclidean distance is the $2$-norm of their difference. A line $ax+by+c=0$ is a level set of a linear function, so the distance from a point to the line is that function's value divided by the norm of its gradient. Conics are the quadratic curves; completing the square on $x^2+y^2+Dx+Ey+F=0$ yields a circle; $e = c/a$ classifies ellipse ($e<1$), parabola ($e=1$) and hyperbola ($e>1$). Polar coordinates $(r,\theta)$ are the Euclidean plane written in rotationally natural coordinates.

Dimensional check. The declared output unit is degree; ToolBox evaluates the formula in SI (radians internally for every trigonometric call) and python-calc converts to the unit on the card.

History and Development

• Descartes, La Géométrie (1637), and Fermat. Coordinates as a method; the name “Cartesian.”

• Apollonius of Perga (3rd century BCE). Conics — ellipse, parabola, hyperbola — centuries before coordinates, in purely geometric language.

• Euler, Lagrange. Rotation of axes to eliminate $xy$ terms; classification of the general conic.

• 19th century. Vector geometry (Gibbs, Heaviside) rewrites point-to-plane and skew-line distance as $|(\mathbf{P}_2-\mathbf{P}_1)\cdot(\mathbf{d}_1\times\mathbf{d}_2)|/|\mathbf{d}_1\times\mathbf{d}_2|$.

Related Concepts: Distance 3D, Slope, Angle Between Two Lines, Point To Line Distance, Circle Radius From General Form, Parabola Latus Rectum, Ellipse Eccentricity, Ellipse Linear Eccentricity