Point coordinates: \(x,\) \(y,\) \(z,\) \({x_0},\) \({y_0},\) \({z_0},\) \({x_1},\) \({y_1},\) \({z_1},\ldots\)

Real numbers: \(A,\) \(B,\) \(C,\) \(D,\) \({A_1},\) \({B_1},\) \(\ldots,\) \(a,\) \(b,\) \(c,\) \({a_1},\) \({b_1},\) \(\ldots,\) \(\lambda,\) \(p,\) \(t,\) \(s,\) \(\ldots\)

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: \(d\)

Direction cosines: \(\cos\alpha\), \(\cos\beta\), \(\cos\gamma\)

Distance from a point to a plane: \(d\)

- The general equation of a plane in the Cartesian coordinate system is represented by the linear equation

\(Ax + By + Cz \) \(+\,D =0.\) - 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

\(Ax + By + Cz \) \(+\, D =0.\) - 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. - 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. - 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. - 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. - 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. - 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, \(s\) and \(t\) are parameters, 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. - 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

\(\cos \varphi = {\large\frac{{{\mathbf{n_1}} \cdot {\mathbf{n_2}}}}{{\left| {{\mathbf{n_1}}} \right| \cdot \left| {{\mathbf{n_2}}} \right|}}\normalsize} =\) \( {\large\frac{{{A_1}{A_2} + {B_1}{B_2} + {C_1}{C_2}}}{{\sqrt {A_1^2 + B_1^2 + C_1^2} \sqrt {A_2^2 + B_2^2 + C_2^2} }}\normalsize}.\) - 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

\({\large\frac{{{A_1}}}{{{A_2}}}\normalsize} = {\large\frac{{{B_1}}}{{{B_2}}}\normalsize} = {\large\frac{{{C_1}}}{{{C_2}}}\normalsize}.\) - 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

\({A_1}{A_2} + {B_1}{B_2} \) \(+\,{C_1}{C_2} \) \(= 0.\) - 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 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

\(\left| {\begin{array}{*{20}{c}} {x – {x_1}} & {y – {y_1}} & {z – {z_1}}\\ {{x_2} – {x_1}} & {{y_2} – {y_1}} & {{z_2} – {z_1}}\\ a & b & c \end{array}} \right| \) \(= 0.\) - 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

\(d = \left| {\large\frac{{A{x_1} + B{y_1} + C{z_1} + D}}{{\sqrt {{A^2} + {B^2} + {C^2}} }}\normalsize} \right|.\) - 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

\(a = \left| {\begin{array}{*{20}{c}} {{B_1}} & {{C_1}}\\ {{B_2}} & {{C_2}} \end{array}} \right|,\;\) \(b = \left| {\begin{array}{*{20}{c}} {{C_1}} & {{A_1}}\\ {{C_2}} & {{A_2}} \end{array}} \right|,\;\) \(c = \left| {\begin{array}{*{20}{c}} {{A_1}} & {{B_1}}\\ {{A_2}} & {{B_2}} \end{array}} \right|,\)

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