Formulas and Tables

Analytic Geometry

Plane

Point coordinates: \(x,\) \(y,\) \(z,\) \({x_0},\) \({y_0},\) \({z_0},\) \({x_1},\) \({y_1},\) \({z_1},\ldots\)
Real numbers: \(A,\) \(B,\) \(C,\) \(D,\) \({A_1},\) \({B_1},\) \(\ldots,\) \(a,\) \(b,\) \(c,\) \({a_1},\) \({b_1},\) \(\ldots,\) \(\lambda,\) \(p,\) \(t,\) \(s,\) \(\ldots\)

Normal vectors: \(\mathbf{n}\), \(\mathbf{n_1}\), \(\mathbf{n_2}\)
Direction cosines: \(\cos\alpha\), \(\cos\beta\), \(\cos\gamma\)
Distance from a point to a plane: \(d\)

  1. The general equation of a plane in the Cartesian coordinate system is represented by the linear equation
    \(Ax + By + Cz \) \(+\,D =0.\)
  2. The coordinates of the normal vector \(\mathbf{n}\left( {A,B,C} \right)\) to a plane are the coefficients in the general equation of the plane
    \(Ax + By + Cz \) \(+\, D =0.\)
Normal vector to a plane
  1. Special cases of the equation of a plane
    \(Ax + By + Cz \) \(+\, D =0\)

    If \(A = 0\), the plane is parallel to the \(x\)-axis;
    If \(B = 0\), the plane is parallel to the \(y\)-axis;
    If \(C = 0\), the plane is parallel to the \(z\)-axis;
    If \(D = 0\), the plane passes through the origin.

    If \(A = B = 0\), the plane is parallel to the \(xy\)-plane;
    If \(B = C = 0\), the plane is parallel to the \(yz\)-plane;
    If \(A = C = 0\), the plane is parallel to the \(xz\)-plane.

  2. Point direction form of a plane equation
    \(A\left( {x – {x_0}} \right) \) \(+\, B\left( {y – {y_0}} \right) \) \(+\, C\left( {z – {z_0}} \right) \) \(= 0,\) where the point \(P\left( {{x_0},{y_0},{z_0}} \right)\) lies in the plane and the vector \(\mathbf{n}\left( {A,B,C} \right)\) is normal to the plane.
Point direction form of a plane equation
  1. Intercept form of a plane equation
    \({\large\frac{x}{a}\normalsize} + {\large\frac{y}{b}\normalsize} + {\large\frac{z}{c}\normalsize} = 1,\) where \(a,\) \(b,\) \(c\) are the intercepts on the \(x,\) \(y\) and \(z\) axes, respectively.
Intercept form of a plane equation
  1. Three points form of a plane equation
    \(\left| {\begin{array}{*{20}{c}}
    {x – {x_3}} & {y – {y_3}} & {z – {z_3}}\\
    {{x_1} – {x_3}} & {{y_1} – {y_3}} & {{z_1} – {z_3}}\\
    {{x_2} – {x_3}} & {{y_2} – {y_3}} & {{z_2} – {z_3}}
    \end{array}} \right| \) \(= 0\;\) or \(\left| {\begin{array}{*{20}{l}}
    x & y & z & 1\\
    {{x_1}} & {{y_1}} & {{z_1}} & 1\\
    {{x_2}} & {{y_2}} & {{z_2}} & 1\\
    {{x_3}} & {{y_3}} & {{z_3}} & 1
    \end{array}} \right| = 0,\)
    where the points \(A\left( {{x_1},{y_1},{z_1}} \right),\) \(B\left( {{x_2},{y_2},{z_2}} \right),\) \(C\left( {{x_3},{y_3},{z_3}} \right)\) lie in the given plane.
Three points form of a plane equation
  1. Normal form of a plane equation
    \(x\cos \alpha + y\cos \beta \) \(+\, z\cos \gamma \) \(-\;p \) \(= 0\)
    Here \(p\) is the distance from the origin to the plane, and \(\cos \alpha,\) \(\cos \beta,\) \(\cos \gamma\) are the direction cosines of any straight line normal to the plane.
Normal form of a plane equation
  1. Equation of a plane in parametric form
    \(\left\{
    \begin{aligned}
    x &= {x_1} + {a_1}s + {a_2}t \\
    y &= {y_1} + {b_1}s + {b_2}t \\
    z &= {z_1} + {c_1}s + {c_2}t
    \end{aligned}
    \right.,\)
    where \(\left( {x,y,z} \right)\) are the coordinates of any point of the plane, the point \(P\left( {{x_1},{y_1},{z_1}} \right)\) lies in this plane, and the vectors \(\mathbf{u}\left( {{a_1},{b_1},{c_1}} \right),\) \(\mathbf{v}\left( {{a_2},{b_2},{c_2}} \right)\) are parallel to the plane.
Equation of a plane in parametric form
  1. Dihedral angle between two planes
    Let the planes be given by the equations
    \({A_1}x + {B_1}y + {C_1}z \) \(+\,{D_1} \) \(= 0,\)
    \({A_2}x + {B_2}y + {C_2}z \) \(+\, {D_2} \) \(= 0.\)
    Then the dihedral angle between them is expressed by the formula \(\cos \varphi = {\large\frac{{{\mathbf{n_1}} \cdot {\mathbf{n_2}}}}{{\left| {{\mathbf{n_1}}} \right| \cdot \left| {{\mathbf{n_2}}} \right|}}\normalsize} =\) \( {\large\frac{{{A_1}{A_2} + {B_1}{B_2} + {C_1}{C_2}}}{{\sqrt {A_1^2 + B_1^2 + C_1^2} \sqrt {A_2^2 + B_2^2 + C_2^2} }}\normalsize}.\)
Dihedral angle between two planes
  1. Parallel planes
    Two planes \({A_1}x + {B_1}y + {C_1}z \) \(+\,{D_1} \) \(= 0\) and \({A_2}x + {B_2}y + {C_2}z \) \(+\,{D_2} \) \(= 0\) are parallel if
    \({\large\frac{{{A_1}}}{{{A_2}}}\normalsize} = {\large\frac{{{B_1}}}{{{B_2}}}\normalsize} = {\large\frac{{{C_1}}}{{{C_2}}}\normalsize}.\)
  2. Perpendicular planes
    Two planes \({A_1}x + {B_1}y + {C_1}z \) \(+\,{D_1} \) \(= 0\) and \({A_2}x + {B_2}y + {C_2}z \) \(+\,{D_2} \) \(= 0\) are perpendicular if
    \({A_1}{A_2} + {B_1}{B_2} \) \(+\,{C_1}{C_2} \) \(= 0.\)
  3. Equation of a plane given a point and two vectors
    The plane passing through the point \(P\left( {{x_1},{y_1},{z_1}} \right)\) and parallel to two non-collinear vectors \(\mathbf{u}\left( {{a_1},{b_1},{c_1}} \right)\) and \(\mathbf{v}\left( {{a_2},{b_2},{c_2}} \right)\) is determined by the equation
    \(\left| {\begin{array}{*{20}{c}}
    {x – {x_1}} & {y – {y_1}} & {z – {z_1}}\\
    {{a_1}} & {{b_1}} & {{c_1}}\\
    {{a_2}} & {{b_2}} & {{c_2}}
    \end{array}} \right| \) \(= 0.\)
Equation of a plane given a point and two vectors
  1. Equation of a plane given two points and a vector
    The plane passing through the points \({P_1}\left( {{x_1},{y_1},{z_1}} \right)\) and \({P_2}\left( {{x_2},{y_2},{z_2}} \right)\) and parallel to the vector \(\mathbf{u}\left( {a,b,c} \right)\) is described by the equation
    \(\left| {\begin{array}{*{20}{c}}
    {x – {x_1}} & {y – {y_1}} & {z – {z_1}}\\
    {{x_2} – {x_1}} & {{y_2} – {y_1}} & {{z_2} – {z_1}}\\
    a & b & c
    \end{array}} \right| \) \(= 0.\)
Equation of a plane given two points and a vector
  1. Distance from a point to a plane
    The distance from the point \({P_1}\left( {{x_1},{y_1},{z_1}} \right)\) to the plane \(Ax + By + Cz \) \(+\,D \) \(= 0\) is determined by the formula
    \(d = \left| {\large\frac{{A{x_1} + B{y_1} + C{z_1} + D}}{{\sqrt {{A^2} + {B^2} + {C^2}} }}\normalsize} \right|.\)
Distance from a point to a plane
  1. Intersection of two planes
    If two planes \({A_1}x + {B_1}y + {C_1}z \) \(+\,{D_1} \) \(= 0\) and \({A_2}x + {B_2}y + {C_2}z \) \(+\, {D_2} \) \(= 0\) intersect, the intersection straight line is given by the equation
    \(
    \left\{
    \begin{aligned}
    x &= {x_1} + at \\
    y &= {y_1} + bt \\
    z &= {z_1} + ct
    \end{aligned}
    \right.\;\)  or  \({\large\frac{{x – {x_1}}}{a}\normalsize} = {\large\frac{{y – {y_1}}}{b}\normalsize} =\) \( {\large\frac{{z – {z_1}}}{c}\normalsize},\)
    where
    \(a = \left| {\begin{array}{*{20}{c}}
    {{B_1}} & {{C_1}}\\
    {{B_2}} & {{C_2}}
    \end{array}} \right|,\;\) \(b = \left| {\begin{array}{*{20}{c}}
    {{C_1}} & {{A_1}}\\
    {{C_2}} & {{A_2}}
    \end{array}} \right|,\;\) \(c = \left| {\begin{array}{*{20}{c}}
    {{A_1}} & {{B_1}}\\
    {{A_2}} & {{B_2}}
    \end{array}} \right|,\)

    \({x_1} = {\large\frac{{b\left| {\begin{array}{*{20}{c}}
    {{D_1}}&{{C_1}}\\
    {{D_2}}&{{C_2}}
    \end{array}} \right| – c\left| {\begin{array}{*{20}{c}}
    {{D_1}}&{{B_1}}\\
    {{D_2}}&{{B_2}}
    \end{array}} \right|}}{{{a^2} + {b^2} + {c^2}}}\normalsize},\;\) \({y_1} = {\large\frac{{c\left| {\begin{array}{*{20}{c}}
    {{D_1}}&{{A_1}}\\
    {{D_2}}&{{A_2}}
    \end{array}} \right| – a\left| {\begin{array}{*{20}{c}}
    {{D_1}}&{{C_1}}\\
    {{D_2}}&{{C_2}}
    \end{array}} \right|}}{{{a^2} + {b^2} + {c^2}}}\normalsize},\;\) \({z_1} = {\large\frac{{a\left| {\begin{array}{*{20}{c}}
    {{D_1}}&{{B_1}}\\
    {{D_2}}&{{B_2}}
    \end{array}} \right| – b\left| {\begin{array}{*{20}{c}}
    {{D_1}}&{{A_1}}\\
    {{D_2}}&{{A_2}}
    \end{array}} \right|}}{{{a^2} + {b^2} + {c^2}}}\normalsize.}\)