3D Analytical Geometry: Formulas for Lines and Planes

Relative Positions in 3D Space

Position of Three Planes

The relative position of three planes is determined by analyzing the rank of the system of equations.

• Intersect at one point: The rank of the coefficient matrix and the augmented matrix is 3.
• Intersect in a line: The rank of both matrices is 2.
• Parallel or Coincident: If the ranks are 1 or 2, the planes can be parallel, coincident, or form a prismatic surface.

Position of Two Lines

Given two lines with direction vectors v₁, v₂ and points P₁, P₂:

• Intersecting Lines: The lines are coplanar. The determinant of the matrix formed by vectors v₁, v₂, and P₁P₂ is zero. To find the intersection point, solve the system of their parametric equations.
• Parallel or Coincident Lines: The direction vectors v₁ and v₂ are proportional.

Position of a Line and a Plane

• If they intersect, there is a single solution point.
• If the line is parallel to the plane, there is no solution.
• If the line is contained within the plane, there are infinite solutions.

To find an intersection point, substitute the line's parametric equations into the plane's equation and solve for the parameter (λ). The resulting value is then substituted back into the line's equations to find the point's coordinates.

Angle Calculations

Angle Between Two Planes

The angle is calculated using the dot product of their normal vectors (n₁ and n₂).

cos(θ) = |n₁ ⋅ n₂| / (|n₁| |n₂|)

Angle Between a Line and a Plane

The angle is calculated using the dot product of the plane's normal vector (n) and the line's direction vector (u).

sin(θ) = |n ⋅ u| / (|n| |u|)

Angle Between Two Lines

The angle is calculated using the dot product of their direction vectors (v₁ and v₂).

cos(θ) = |v₁ ⋅ v₂| / (|v₁| |v₂|)

Distance Calculations

Distance Between Two Points

For points P(x₁, y₁, z₁) and Q(x₂, y₂, z₂):

d(P, Q) = √((x₂ - x₁)² + (y₂ - y₁)² + (z₂ - z₁)² )

Distance from a Point to a Plane

For a point P(x₀, y₀, z₀) and a plane Ax + By + Cz + D = 0:

d(P, π) = |Ax₀ + By₀ + Cz₀ + D| / √(A² + B² + C²)

Distance Between Two Parallel Planes

Take any point from one plane and calculate its distance to the other plane using the formula above.

Distance from a Point to a Line

For a point P and a line defined by point A and direction vector u:

d(P, r) = |AP x u| / |u|

Distance Between Two Skew Lines

For lines r₁(P₁, v₁) and r₂(P₂, v₂), the distance is the absolute value of the scalar triple product of the vector connecting their points (P₁P₂) and their direction vectors, divided by the magnitude of the cross product of their direction vectors.

d(r₁, r₂) = |det(P₁P₂, v₁, v₂)| / |v₁ x v₂|

Geometric Formulas

Equation of a Line Through a Point Intersecting Two Lines

To find the equation of a line that passes through a given point and intersects two other lines, find the intersection of the two planes that are each formed by the point and one of the lines.

Area of a Triangle

Given vertices A, B, and C, the area is half the magnitude of the cross product of two vectors forming the sides.

Area = (1/2) * |AB x AC|

Volume of a Tetrahedron

Given vertices A, B, C, and D, the volume is one-sixth of the absolute value of the scalar triple product of the vectors forming three concurrent edges.

Volume = (1/6) * |det(AB, AC, AD)|