Parallel planes

Content

Classical Definition

Two planes are called parallel if they do not have common points. (Sometimes coinciding planes are also considered parallel, which simplifies the formulation of some theorems)

The properties

If two parallel planes are crossed by a third, then their intersection lines are parallel
Through a point outside this plane, you can draw a plane parallel to this, and moreover, only one
The segments of parallel lines bounded by two parallel planes are equal
Two angles with respectively parallel and equally directed sides are equal and lie in parallel planes

Sign

If the plane α is parallel to each of two intersecting lines lying in the other plane β, then these planes are parallel

Analytical Definition

If the planes

$A_{1}x+B_{1}y+C_{1}z+D_{1}=0$ ${\ displaystyle A_ {1} x + B_ {1} y + C_ {1} z + D_ {1} = 0}$ and $A_{2}x+B_{2}y+C_{2}z+D_{2}=0$ ${\ displaystyle A_ {2} x + B_ {2} y + C_ {2} z + D_ {2} = 0}$

are parallel then normal vectors $N_{1}(A_{1},B_{1},C_{1})$ ${\ displaystyle N_ {1} (A_ {1}, B_ {1}, C_ {1})}$ and $N_{2}(A_{2},B_{2},C_{2})$ ${\ displaystyle N_ {2} (A_ {2}, B_ {2}, C_ {2})}$ collinear (and vice versa). Therefore, the condition

${\frac {A_{2}}{A_{1}}}={\frac {B_{2}}{B_{1}}}={\frac {C_{2}}{C_{1}}}$ ${\ displaystyle {\ frac {A_ {2}} {A_ {1}}} = {\ frac {B_ {2}} {B_ {1}}} = {\ frac {C_ {2}} {C_ {1 }}}}$ ^[1] is a necessary and sufficient condition for parallelism or coincidence of planes.

Example 1

Planes $2x-3y-4z+11=0$ ${\ displaystyle 2x-3y-4z + 11 = 0}$ and $-4x+6y+8z+36=0$ ${\ displaystyle -4x + 6y + 8z + 36 = 0}$ parallel since ${\frac {-4}{2}}={\frac {6}{-3}}={\frac {8}{-4}}$ ${\ displaystyle {\ frac {-4} {2}} = {\ frac {6} {- 3}} = {\ frac {8} {- 4}}}$

Example 2

Planes $2x-3z-12=0(A_{1}=2,B_{1}=0,C_{1}=-3)$ ${\ displaystyle 2x-3z-12 = 0 (A_ {1} = 2, B_ {1} = 0, C_ {1} = - 3)}$ and $4x+4y-6z+7=0(A_{2}=4,B_{2}=4,C_{2}=-6)$ ${\ displaystyle 4x + 4y-6z + 7 = 0 (A_ {2} = 4, B_ {2} = 4, C_ {2} = - 6)}$ are not parallel since $B_{1}=0$ ${\ displaystyle B_ {1} = 0}$ , but $B_{2}\neq 0$ ${\ displaystyle B_ {2} \ neq 0}$
Remark . If not only the coefficients at the coordinates, but also the free terms are proportional, that is, if
${\frac {A_{2}}{A_{1}}}={\frac {B_{2}}{B_{1}}}={\frac {C_{2}}{C_{1}}}={\frac {D_{2}}{D_{1}}},$ ${\ displaystyle {\ frac {A_ {2}} {A_ {1}}} = {\ frac {B_ {2}} {B_ {1}}} = {\ frac {C_ {2}} {C_ {1 }}} = {\ frac {D_ {2}} {D_ {1}}},}$ ^[2] then the planes coincide. So equations $3x+7y+5z+4=0$ ${\ displaystyle 3x + 7y + 5z + 4 = 0}$ and $6x+14y+10z+8=0$ ${\ displaystyle 6x + 14y + 10z + 8 = 0}$ represent the same plane.

Notes

↑ at $A_{1},B_{1},C_{1}\neq 0$ ${\ displaystyle A_ {1}, B_ {1}, C_ {1} \ neq 0}$ . If a $A_{1}=0$ ${\ displaystyle A_ {1} = 0}$ then $A_{2}=0,{\frac {B_{2}}{B_{1}}}={\frac {C_{2}}{C_{1}}}$ ${\ displaystyle A_ {2} = 0, {\ frac {B_ {2}} {B_ {1}}} = {\ frac {C_ {2}} {C_ {1}}}}$ . Similarly for $B_{1}=0$ ${\ displaystyle B_ {1} = 0}$ or $C_{1}=0$ ${\ displaystyle C_ {1} = 0}$ .
↑ at $A_{1},B_{1},C_{1},D_{1}\neq 0$ ${\ displaystyle A_ {1}, B_ {1}, C_ {1}, D_ {1} \ neq 0}$ . If a $A_{1}=0$ ${\ displaystyle A_ {1} = 0}$ then $A_{2}=0,{\frac {B_{2}}{B_{1}}}={\frac {C_{2}}{C_{1}}}={\frac {D_{2}}{D_{1}}}$ ${\ displaystyle A_ {2} = 0, {\ frac {B_ {2}} {B_ {1}}} = {\ frac {C_ {2}} {C_ {1}}} = {\ frac {D_ { 2}} {D_ {1}}}}$ . Similarly for $B_{1}=0,C_{1}=0$ ${\ displaystyle B_ {1} = 0, C_ {1} = 0}$ or $D_{1}=0$ ${\ displaystyle D_ {1} = 0}$ .

[1] t $A_{1},B_{1},C_{1}\neq 0$ ${\ displaystyle A_ {1}, B_ {1}, C_ {1} \ neq 0}$ . If a $A_{1}=0$ ${\ displaystyle A_ {1} = 0}$ then $A_{2}=0,{\frac {B_{2}}{B_{1}}}={\frac {C_{2}}{C_{1}}}$ ${\ displaystyle A_ {2} = 0, {\ frac {B_ {2}} {B_ {1}}} = {\ frac {C_ {2}} {C_ {1}}}}$ . Similarly for $B_{1}=0$ ${\ displaystyle B_ {1} = 0}$ or $C_{1}=0$ ${\ displaystyle C_ {1} = 0}$ .

[2] t $A_{1},B_{1},C_{1},D_{1}\neq 0$ ${\ displaystyle A_ {1}, B_ {1}, C_ {1}, D_ {1} \ neq 0}$ . If a $A_{1}=0$ ${\ displaystyle A_ {1} = 0}$ then $A_{2}=0,{\frac {B_{2}}{B_{1}}}={\frac {C_{2}}{C_{1}}}={\frac {D_{2}}{D_{1}}}$ ${\ displaystyle A_ {2} = 0, {\ frac {B_ {2}} {B_ {1}}} = {\ frac {C_ {2}} {C_ {1}}} = {\ frac {D_ { 2}} {D_ {1}}}}$ . Similarly for $B_{1}=0,C_{1}=0$ ${\ displaystyle B_ {1} = 0, C_ {1} = 0}$ or $D_{1}=0$ ${\ displaystyle D_ {1} = 0}$ .