# 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
$$Ax + By + Cz$$ $$+\, D =0.$$
3. 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.
4. 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.
5. 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.
6. 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.
7. 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.
8. 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.
9. 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
$$\cos \varphi = {\large\frac{{{\mathbf{n_1}} \cdot {\mathbf{n_2}}}}{{\left| {{\mathbf{n_1}}} \right| \cdot \left| {{\mathbf{n_2}}} \right|}}\normalsize} =$$ $${\large\frac{{{A_1}{A_2} + {B_1}{B_2} + {C_1}{C_2}}}{{\sqrt {A_1^2 + B_1^2 + C_1^2} \sqrt {A_2^2 + B_2^2 + C_2^2} }}\normalsize}.$$
10. 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
$${\large\frac{{{A_1}}}{{{A_2}}}\normalsize} = {\large\frac{{{B_1}}}{{{B_2}}}\normalsize} = {\large\frac{{{C_1}}}{{{C_2}}}\normalsize}.$$
11. 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
$${A_1}{A_2} + {B_1}{B_2}$$ $$+\,{C_1}{C_2}$$ $$= 0.$$
12. 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
$$\left| {\begin{array}{*{20}{c}} {x – {x_1}} & {y – {y_1}} & {z – {z_1}}\\ {{a_1}} & {{b_1}} & {{c_1}}\\ {{a_2}} & {{b_2}} & {{c_2}} \end{array}} \right|$$ $$= 0.$$
13. 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
$$\left| {\begin{array}{*{20}{c}} {x – {x_1}} & {y – {y_1}} & {z – {z_1}}\\ {{x_2} – {x_1}} & {{y_2} – {y_1}} & {{z_2} – {z_1}}\\ a & b & c \end{array}} \right|$$ $$= 0.$$
14. 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
$$d = \left| {\large\frac{{A{x_1} + B{y_1} + C{z_1} + D}}{{\sqrt {{A^2} + {B^2} + {C^2}} }}\normalsize} \right|.$$
15. 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
$$a = \left| {\begin{array}{*{20}{c}} {{B_1}} & {{C_1}}\\ {{B_2}} & {{C_2}} \end{array}} \right|,\;$$ $$b = \left| {\begin{array}{*{20}{c}} {{C_1}} & {{A_1}}\\ {{C_2}} & {{A_2}} \end{array}} \right|,\;$$ $$c = \left| {\begin{array}{*{20}{c}} {{A_1}} & {{B_1}}\\ {{A_2}} & {{B_2}} \end{array}} \right|,$$
$${x_1} = {\large\frac{{b\left| {\begin{array}{*{20}{c}} {{D_1}}&{{C_1}}\\ {{D_2}}&{{C_2}} \end{array}} \right| – c\left| {\begin{array}{*{20}{c}} {{D_1}}&{{B_1}}\\ {{D_2}}&{{B_2}} \end{array}} \right|}}{{{a^2} + {b^2} + {c^2}}}\normalsize},\;$$ $${y_1} = {\large\frac{{c\left| {\begin{array}{*{20}{c}} {{D_1}}&{{A_1}}\\ {{D_2}}&{{A_2}} \end{array}} \right| – a\left| {\begin{array}{*{20}{c}} {{D_1}}&{{C_1}}\\ {{D_2}}&{{C_2}} \end{array}} \right|}}{{{a^2} + {b^2} + {c^2}}}\normalsize},\;$$ $${z_1} = {\large\frac{{a\left| {\begin{array}{*{20}{c}} {{D_1}}&{{B_1}}\\ {{D_2}}&{{B_2}} \end{array}} \right| – b\left| {\begin{array}{*{20}{c}} {{D_1}}&{{A_1}}\\ {{D_2}}&{{A_2}} \end{array}} \right|}}{{{a^2} + {b^2} + {c^2}}}\normalsize.}$$