The Plane in Three Dimensions: Equations, Dihedral Angles & Distance Metrics

1. General First-Degree Equation in $\mathbb{R}^3$

Proof: Let $P_1(x_1, y_1, z_1)$ and $P_2(x_2, y_2, z_2)$ be any two distinct points on the surface $Ax + By + Cz + D = 0$. Consider any point $P$ dividing $P_1 P_2$ in ratio $k : 1$:

Substituting into the equation:

Since every point on the line segment $P_1 P_2$ lies entirely on the surface, the surface is a plane. $\blacksquare$

2. Geometric Meaning of Coefficients: The Normal Vector $\vec{n}$

Let $P_1(x_1, y_1, z_1)$ and $P_2(x_2, y_2, z_2)$ lie on the plane. Subtracting their equations:

The vector $\vec{n} = A\hat{i} + B\hat{j} + C\hat{k}$ is perpendicular to every displacement vector lying in the plane, establishing that $(A, B, C)$ are the direction ratios of the normal vector to the plane.

3. Intercept Form of the Plane

If the plane intersects the coordinate axes at $A(a, 0, 0)$, $B(0, b, 0)$, and $C(0, 0, c)$ ($abc \ne 0$), dividing by $-D$ yields:

1. The Hesse Normal Form: $lx + my + nz = p$

Let $ON$ be the perpendicular drawn from origin $O$ to the plane, of length $p \ge 0$, having direction cosines $(l, m, n)$. For any point $P(x, y, z)$ on the plane, the projection of position vector $\vec{OP} = (x, y, z)$ onto the normal unit vector $\hat{n} = (l, m, n)$ is identically $p$:

2. Reduction of General Equation to Normal Form

To convert $Ax + By + Cz + D = 0$ into normal form, transpose $D$ so the RHS is non-negative and divide by $\pm\sqrt{A^2 + B^2 + C^2}$:

The sign is chosen opposite to $D$ so the constant distance $p$ is positive.

3. Plane Passing Through Three Non-Collinear Points

The plane passing through $P_1(x_1, y_1, z_1)$, $P_2(x_2, y_2, z_2)$, and $P_3(x_3, y_3, z_3)$ is given by the coplanarity determinant:

1. Dihedral Angle Between Two Planes

The angle $\theta$ between two planes $\Pi_1: A_1 x + B_1 y + C_1 z + D_1 = 0$ and $\Pi_2: A_2 x + B_2 y + C_2 z + D_2 = 0$ is equal to the angle between their normal vectors $\vec{n}_1$ and $\vec{n}_2$:

2. Special Geometric Orientations

• Perpendicular Planes ($\Pi_1 \perp \Pi_2$): $$\mathbf{A_1 A_2 + B_1 B_2 + C_1 C_2 = 0}$$
• Parallel Planes ($\Pi_1 \parallel \Pi_2$): $$\mathbf{\frac{A_1}{A_2} = \frac{B_1}{B_2} = \frac{C_1}{C_2} \ne \frac{D_1}{D_2}}$$
• Coincident Planes: The ratios of all four coefficients are identical.

1. Distance from a Point $P_1(x_1, y_1, z_1)$ to a Plane

Let the plane be $\Pi: Ax + By + Cz + D = 0$. The perpendicular distance $d$ from $P_1$ to $\Pi$ is given by:

Proof: Translate the coordinate origin to $P_1(x_1, y_1, z_1)$ via $x = x' + x_1, y = y' + y_1, z = z' + z_1$. The equation becomes $Ax' + By' + Cz' + (Ax_1 + By_1 + Cz_1 + D) = 0$. In this frame, the distance from the new origin $(0, 0, 0)$ is the constant term divided by the normal norm. $\blacksquare$

2. Distance Between Parallel Planes

For parallel planes $Ax + By + Cz + D_1 = 0$ and $Ax + By + Cz + D_2 = 0$:

3. Equations of the Dihedral Bisector Planes

The locus of points equidistant from two planes $\Pi_1$ and $\Pi_2$ forms two orthogonal bisector planes:

To distinguish the acute from the obtuse bisector: make $D_1, D_2 > 0$. If $A_1 A_2 + B_1 B_2 + C_1 C_2 > 0$, the positive sign gives the obtuse bisector and the negative sign gives the acute bisector.

1. Family (Pencil) of Planes Passing Through the Intersection of Two Planes

Let $\Pi_1: A_1 x + B_1 y + C_1 z + D_1 = 0$ and $\Pi_2: A_2 x + B_2 y + C_2 z + D_2 = 0$ be two non-parallel planes. Their intersection is a straight line $L$. Any plane passing through this line $L$ is represented by the linear combination parameter $\lambda \in \mathbb{R}$:

2. Geometric Determination of the Parameter $\lambda$

The parameter $\lambda$ is uniquely fixed by imposing an extra geometric constraint, such as:

1. Passing through a fourth point $P_0(x_0, y_0, z_0)$.
2. Being perpendicular to a third plane $\Pi_3: A_3 x + B_3 y + C_3 z + D_3 = 0$ via $(A_1 + \lambda A_2)A_3 + (B_1 + \lambda B_2)B_3 + (C_1 + \lambda C_2)C_3 = 0$.
3. Being parallel to a given line with direction ratios $(l, m, n)$.