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Derivation/Representation of Equation of Planes

Planes are best represented by equations in Cartesian Coordinate System. As with any geometic object in any coordinate system, equations of Planes can be represented in Scalar Coordinate form, Scalar Parametric form, Vector Parametric form and Vector Non-Parametric form.

Equations in Scalar Coordinate Form

Equations of Planes in Scalar Form are very similar to the Equations of 2D Lines in Scalar Form

  1. Standard Form or Implicit Form: The equation of the plane in Standard Form or Implicit Form is given as:

    \(Ax + By + Cz + D=0\)

    Please note that this equation is similar to the Standard Form of Line Equation in 2D. Any equation of plane derived from any given inputs is generally brought into the Standard Form for final representation. Following are certain important observations abount scalar equations of Planes:
    1. It is represented by a single Linear Equation in either 1, 2 or 3 variables. The variables are denoted by the coordinates \(x\) and/or \(y\) and/or \(z\)
    2. The co-efficients of \(x\), \(y\) and \(z\) (given as \(A\), \(B\) and \(C\) repectively in the above equation) cannot all be simultaneously zero. Though any 2 can be. If the co-efficient of \(x\) is 0 (i.e. \(A=0\)) then the plane is either parallel to or contains X axis. If the co-efficient of \(y\) is 0 (i.e. \(B=0\)) then the plane is either parallel to or contains Y axis. If the co-efficient of \(z\) is 0 (i.e. \(C=0\)) then the plane is either parallel to or contains Z axis. If any 2 of the co-efficients are zero, the plane is either parallel to or contains those axes and is perpendicular to the non-zero co-efficient axis
    3. \(D\) is the constant of the equation. If \(D=0\), the plane passes through origin.
  2. Normal Form: The Normal Form Equation of a Plane passing through point \((x_1,y_1,z_1)\) having a normal vector with direction ratio \((x_n,y_n,z_n)\) is given as follows:

    \(x_n(x-x_1) + y_n(y-y_1) + z_n(z-z_1) = 0\)

    \(\Rightarrow x_nx + y_ny + z_nz -x_nx_1 - y_ny_1 - z_nz_1 =0 \)

    Please note that this equation is similar to the Normal Form of Line Equation in 2D. Simplyfying the above equation we get an equation of the Standard Form or Implicit Form of plane

    \(Ax + By + Cz + D=0\)

    In the above equation

    \(A = x_n\)
    \(B = y_n\)
    \(C = z_n\)
    \(D = -x_nx_1 - y_ny_1 - z_nz_1\)
  3. Unit Normal Distance Form: The Standard Form of plane equation is given as:

    \(Ax + By + Cz + D =0\)

    The Unit Normal Distance Form of Line is given by dividing both sides by \(\sqrt{A^2 + B^2 + C^2}\) and rearranging as following:

    \(\frac{A}{\sqrt{A^2 + B^2 + C^2}}x + \frac{B}{\sqrt{A^2 + B^2 + C^2}}y + \frac{C}{\sqrt{A^2 + B^2 + C^2}}z =\frac{-D}{\sqrt{A^2 + B^2 + C^2}}\)

    In this form of equation, the term \(\frac{-D}{\sqrt{A^2 + B^2 + C^2}}\) represents the signed distance of plane from the origin (0,0).
  4. Intercept Form: The Standard Form of plane equation is given as:

    \(Ax + By + Cz + D=0\)
    \(\Rightarrow Ax + By + Cz=-D\)

    Dividing by -D on both sides we get

    \(\frac{x}{\frac{-D}{A}} + \frac{y}{\frac{-D}{B}} + \frac{z}{\frac{-D}{C}} = 1\)

    In the above equation the value \(\frac{-D}{A}\) is known as X Intercept (the \(x\) coordinate of the point where the plane meets the X axis).
    The value \(\frac{-D}{B}\) is known as Y Intercept (the \(y\) coordinate of the point where the plane meets the Y axis).
    The value \(\frac{-D}{C}\) is known as Z Intercept (the \(z\) coordinate of the point where the plane meets the Z axis).
    Hence the equation of the plane having intercepts \(c_x\), \(c_y\) and , \(c_z\) on X, Y and Z axis respectively is given as:

    \(\frac{x}{c_x} + \frac{y}{c_y} + \frac{z}{c_z} = 1 \)

  5. Determinant Form: The Determinant equation of the plane that passes through 3 points \((x_1, y_1, z_1)\), \((x_2, y_2, z_2)\) and \((x_3, y_3, z_3)\) is given as:

    \(\begin{vmatrix} x & y & z & 1\\ x_1 & y_1 & z_1 & 1 \\ x_2& y_2 & z_2 & 1 \\ x_3 & y_3 & z_3 & 1\end{vmatrix}=0\)   OR   \(\begin{vmatrix} (x - x_1) & (y - y_1) & (z - z_1)\\ (x_1 - x_2) & (y_1 - y_2) & (z_1 - z_2)\\ (x_1 - x_3) & (y_1 - y_3) & (z_1 - z_3)\end{vmatrix}=0\)

    The above equation can be brought to Standard Form of plane equation by solving the determinant and rearranging the terms.
    Any of the coordinate points can be subtracted from \(x\) (Only that corresponding coordinate point must be subtracted from \(y\) and \(z\).). Similar consistency must be maintained while subtracting the coordinate points.
  6. Explicit Form: The Explicit Form of plane equations are given as:

    \(x= Ay + Bz + C\)   (When \(x\) is given as a function of \(y\) and \(z\))

    \(y= Az + Bx + C\)   (When \(y\) is given as a function of \(z\) and \(x\))

    \(z= Ax + By + C\)   (When \(z\) is given as a function of \(x\) and \(y\))

    7. Equation in Scalar Parametric Form

    The following are Parametric Form representation of equations of Planes:

    1. Point Direction Ratio Form:The Parametric Equation of a Plane passing through point \((x_1,y_1,z_1)\) containing 2 non parallel vectors having direction ratios \((x_s,y_s,z_s)\) and \((x_d,y_d,z_d)\) is given as follows:

      \(x= x_1 + tx_s + ux_d ,\hspace{.5cm}y= y_1 + ty_s + uy_d ,\hspace{.5cm}z= z_1 + tz_s + uz_d\)

      \(t\) and \(u\) can be any real values.
    2. Explicit Form: The Explicit Parametric Form of plane equations correspond to the Explicit Scalar Form and are given as follows:

      \(y= t ,\hspace{.5cm}z= u ,\hspace{.5cm}x= At + Bu + C\)   (When \(x\) is given as a function of \(y\) and \(z\))

      \(y= t ,\hspace{.5cm}z= u ,\hspace{.5cm}x= Au + C\)   (Plane parallel to \(y\) axis when \(x\) is given as a function of \(z\))

      \(y= t ,\hspace{.5cm}z= Au + C ,\hspace{.5cm}x= u\)   (Plane parallel to \(y\) axis when \(z\) is given as a function of \(x\))

      \(z= t ,\hspace{.5cm}x= u ,\hspace{.5cm}y= At + Bu + C\)   (When \(y\) is given as a function of \(z\) and \(x\))

      \(z= t ,\hspace{.5cm}x= u ,\hspace{.5cm}y= Au + C\)   (Plane parallel to \(z\) axis when \(y\) is given as a function of \(x\))

      \(z= t ,\hspace{.5cm}x= Au + C ,\hspace{.5cm}y= u\)   (Plane parallel to \(z\) axis when \(x\) is given as a function of \(y\))

      \(x= t ,\hspace{.5cm}y= u ,\hspace{.5cm}z= At + Bu + C\)   (When \(z\) is given as a function of \(x\) and \(y\))

      \(x= t ,\hspace{.5cm}y= u ,\hspace{.5cm}z= Au + C\)   (Plane parallel to \(x\) axis when \(z\) is given as a function of \(y\))

      \(x= t ,\hspace{.5cm}y= Au + C ,\hspace{.5cm}z= u\)   (Plane parallel to \(x\) axis when \(y\) is given as a function of \(z\))

      \(x= C ,\hspace{.5cm}y= t ,\hspace{.5cm}z= u\)   (Plane perpendicular to \(x\) axis)

      \(y= C ,\hspace{.5cm}z= t ,\hspace{.5cm}x= u\)   (Plane perpendicular to \(y\) axis)

      \(z= C ,\hspace{.5cm}x= t ,\hspace{.5cm}y= u\)   (Plane perpendicular to \(z\) axis)

      \(t\) and \(u\) can be any real values. \(C\) is a constant.

    Equations in Vector Form

    The following are Vector Form representation of equations of Planes:

    1. Position Vector Form: The Position Vector Form of equation of Plane can be found by using the value of \(x\), \(y\) and \(z\) from the Parametric Equations as follows:

      \(\vec{P}= (x_1 + tx_s + ux_d)\hat{\textbf{i}} + (y_1 + ty_s + uy_d)\hat{\textbf{j}} + (z_1 + tz_s + uz_d)\hat{\textbf{k}}\)   (For a Plane passing through point \((x_1,y_1,z_1)\) containing 2 non parallel vectors having direction ratios \((x_s,y_s,z_s)\) and \((x_d,y_d,z_d)\))

      \(\vec{P}= (At + Bu + C)\hat{\textbf{i}} + t\hat{\textbf{j}} + u\hat{\textbf{k}}\)   (When \(x\) is given as a function of \(y\) and \(z\))

      \(\vec{P}= (Au + C)\hat{\textbf{i}} + t\hat{\textbf{j}} + u\hat{\textbf{k}}\)   (Plane parallel to \(y\) axis when \(x\) is given as a function of \(z\))

      \(\vec{P}= u\hat{\textbf{i}} + t\hat{\textbf{j}} + (Au + C)\hat{\textbf{k}}\)   (Plane parallel to \(y\) axis when \(z\) is given as a function of \(x\))

      \(\vec{P}= u\hat{\textbf{i}} + (At + Bu + C)\hat{\textbf{j}} + t\hat{\textbf{k}}\)   (When \(y\) is given as a function of \(z\) and \(x\))

      \(\vec{P}= u\hat{\textbf{i}} + (Au + C)\hat{\textbf{j}} + t\hat{\textbf{k}}\)   (Plane parallel to \(z\) axis when \(y\) is given as a function of \(x\))

      \(\vec{P}= (Au + C)\hat{\textbf{i}} + u\hat{\textbf{j}} + t\hat{\textbf{k}}\)   (Plane parallel to \(z\) axis when \(x\) is given as a function of \(y\))

      \(\vec{P}= t\hat{\textbf{i}} + u\hat{\textbf{j}} + (At + Bu + C)\hat{\textbf{k}}\)   (When \(z\) is given as a function of \(x\) and \(y\))

      \(\vec{P}= t\hat{\textbf{i}} + u\hat{\textbf{j}} + (Au + C)\hat{\textbf{k}}\)   (Plane parallel to \(x\) axis when \(z\) is given as a function of \(y\))

      \(\vec{P}= t\hat{\textbf{i}} + (Au + C)\hat{\textbf{j}} + u\hat{\textbf{k}}\)   (Plane parallel to \(x\) axis when \(y\) is given as a function of \(z\))

      \(\vec{P}= C\hat{\textbf{i}} + t\hat{\textbf{j}} + u\hat{\textbf{k}}\)   (Plane perpendicular to \(x\) axis)

      \(\vec{P}= u\hat{\textbf{i}} + C\hat{\textbf{j}} + t\hat{\textbf{k}}\)   (Plane perpendicular to \(y\) axis)

      \(\vec{P}= t\hat{\textbf{i}} + u\hat{\textbf{j}} + C\hat{\textbf{k}}\)   (Plane perpendicular to \(z\) axis)
    2. Normal Form: The Normal Form of vector equation of Plane is given as:

      \((\vec{r}-\vec{a})\cdot\vec{n}=0\)   ...(1)
      \(\Rightarrow \vec{r}\cdot\vec{n}-\vec{a}\cdot\vec{n}=0\)
      \(\Rightarrow \vec{r}\cdot\vec{n}=\vec{a}\cdot\vec{n}\)
      \(\Rightarrow \vec{r}\cdot\vec{n}=C\)   ...(2)

      The equations (1) & (2) above are the Normal Form of vector equations of Plane. In above equations

      \(\vec{r}\) is the position vector of any arbitrary point on the Plane (for equation (1) & (2)) or a vector from any point Outside the Plane to a point On the Plane (for equation (2) only).
      \(\vec{a}\) is the position vector of a given point on the Plane.
      \(\vec{n}\) is a vector normal to the Plane.
      \(C\) is a Constant of Line equation if \(\vec{r}\) is the position vector of any arbitrary point on the Plane. Otherwise its an equation specific constant.

      Please note that it is same to the Normal Form of Vector Equation of Line in 2D. Also note that this corresponds to the Normal Form of Scalar Equation of Plane.
    3. Unit Normal Distance Form: The Unit Normal Distance Form of vector equation of a Plane is given as:

      \(\vec{r}\cdot\hat{n}=D\)

      In above equation

      \(\vec{r}\) is either the position vector of any arbitrary point On the Plane or a vector from any point Outside the Plane to a point On the Plane.
      \(\hat{n}\) is a unit normal vector to the Plane.
      \(D\) is signed distance of the Plane from the origin if \(\vec{r}\) is the position vector of any arbitrary point on the Plane. Otherwise its signed distance of point Outside the Plane to the Plane.

      Please note that it is same to the Unit Normal Distance Form of Vector Equation of Line in 2D. Also note that the above equations correspond to the Unit Normal Distance Form of Scalar Equation of Plane.

    Related Topics and Calculators
    Intorduction to Planes,    Finding Points on Plane/Intercepts of Plane,    Types of Planes,    Condition for Coplanarity of 4 Points,    Projection of Vector on a Plane,    Angular Normal of a Plane,    Angle Between 2 Planes,    Angle Between a Line and a Plane,    Relation Between a Line and a Plane,    Relation Between 2 Planes,    Relation Between 3 Planes,    Condition for Collinearity and Concurrency of Planes,    Family of Planes
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