Mathematics

Relative positions in ℝ³

Study of relative positions between points, lines and planes in 3D space using matrix rank.

  • Point and line: A point is contained in a line if it satisfies its equation, otherwise it is exterior.

  • Point and plane: A point is contained in a plane if it satisfies its equation, otherwise it is exterior.

  • 3 collinear points A, B, C: If the vectors AB⃗\vec{AB}, and AC⃗\vec{AC} are proportional:
AB⃗=k⋅AC⃗\vec{AB} = k \cdot \vec{AC}
  • Four coplanar points A, B, C, D: The vectors AB⃗\vec{AB}, AC⃗\vec{AC} and AD⃗\vec{AD} are linearly dependent, that is, the determinant formed by the 3 is zero.
Given line rr with direction vector u⃗\vec{u} and point AA, and line ss with direction vector v⃗\vec{v} and point BB. Let the matrices be:
M=(uxuyuzvxvyvz)M∗=(uxuyuzvxvyvzbx−axby−aybz−az)M = \begin{pmatrix} u_x & u_y & u_z \\ v_x & v_y & v_z \end{pmatrix} \quad M^* = \begin{pmatrix} u_x & u_y & u_z \\ v_x & v_y & v_z \\ b_x - a_x & b_y - a_y & b_z - a_z \end{pmatrix}
  • If Rank M=Rank M∗=1\text{Rank } M = \text{Rank } M^* = 1 they are coincident lines.
  • If Rank M=1\text{Rank } M = 1 and Rank M∗=2\text{Rank } M^* = 2 they are parallel lines.
  • If Rank M=Rank M∗=2\text{Rank } M = \text{Rank } M^* = 2 they are intersecting lines.
  • If Rank M=2\text{Rank } M = 2 and Rank M∗=3\text{Rank } M^* = 3 the lines are skew.
Given the line rr in implicit form:
{A1x+B1y+C1z=D1A2x+B2y+C2z=D2\begin{cases} A_1x + B_1y + C_1z = D_1 \\ A_2x + B_2y + C_2z = D_2 \end{cases}
And the plane π\pi in implicit form:
A3x+B3y+C3z=D3A_3x + B_3y + C_3z = D_3
Let the matrices be:
M=(A1B1C1A2B2C2A3B3C3)M∗=(A1B1C1D1A2B2C2D2A3B3C3D3)M = \begin{pmatrix} A_1 & B_1 & C_1 \\ A_2 & B_2 & C_2 \\ A_3 & B_3 & C_3 \end{pmatrix} \quad M^* = \begin{pmatrix} A_1 & B_1 & C_1 & D_1 \\ A_2 & B_2 & C_2 & D_2 \\ A_3 & B_3 & C_3 & D_3 \end{pmatrix}
  • If Rank M=Rank M∗=2\text{Rank } M = \text{Rank } M^* = 2 the line is contained in the plane.
  • If Rank M=2\text{Rank } M = 2 and Rank M∗=3\text{Rank } M^* = 3 the line is parallel to the plane.
  • If Rank M=Rank M∗=3\text{Rank } M = \text{Rank } M^* = 3 the line and plane intersect.
Given the plane π1\pi_1 in implicit form:
A1x+B1y+C1z=D1A_1x + B_1y + C_1z = D_1
And the plane π2\pi_2 in implicit form:
A2x+B2y+C2z=D2A_2x + B_2y + C_2z = D_2
Let the matrices be:
M=(A1B1C1A2B2C2)M∗=(A1B1C1D1A2B2C2D2)M = \begin{pmatrix} A_1 & B_1 & C_1 \\ A_2 & B_2 & C_2 \end{pmatrix} \quad M^* = \begin{pmatrix} A_1 & B_1 & C_1 & D_1 \\ A_2 & B_2 & C_2 & D_2 \end{pmatrix}
  • If Rank M=Rank M∗=1\text{Rank } M = \text{Rank } M^* = 1 Coincident planes.
  • If Rank M=1\text{Rank } M = 1 and Rank M∗=2\text{Rank } M^* = 2 Parallel planes.
  • If Rank M=Rank M∗=2\text{Rank } M = \text{Rank } M^* = 2 Intersecting planes in a line.
Given 3 general equations of the plane;
π1≡A1x+B1y+C1z=D1π2≡A2x+B2y+C2z=D2π3≡A3x+B3y+C3z=D3\begin{aligned} \pi_1 &\equiv A_1x + B_1y + C_1z = D_1 \\ \pi_2 &\equiv A_2x + B_2y + C_2z = D_2 \\ \pi_3 &\equiv A_3x + B_3y + C_3z = D_3 \end{aligned}
we will study the rank of the coefficient matrix MM and augmented matrix M∗M^*
M=(A1B1C1A2B2C2A3B3C3)M∗=(A1B1C1D1A2B2C2D2A3B3C3D3)M = \begin{pmatrix} A_1 & B_1 & C_1 \\ A_2 & B_2 & C_2 \\ A_3 & B_3 & C_3 \end{pmatrix} \quad M^* = \begin{pmatrix} A_1 & B_1 & C_1 & D_1 \\ A_2 & B_2 & C_2 & D_2 \\ A_3 & B_3 & C_3 & D_3 \end{pmatrix}
Rank MRank M*Relative position
113 Coincident planes
3 coincident planes
12Case 1: 2 coincident planes and another parallel
Case 2: 3 parallel planes
3 parallel planes
22Case 1: 2 coincident planes and another intersecting
Case 2: 3 planes intersecting in a line
3 planes intersecting in a line
23Case 1: 2 parallel planes and another intersecting
Case 2: 3 planes intersecting pair-wise
3 planes intersecting pair-wise
33Intersecting planes.
They intersect at a single point
3 planes intersecting in a point