*Notes to a video lecture on http://www.unizor.com*

__Geometry+ GeoTheorem 1__

*Theorem*

Prove that

**non-null**vector

*(*

**n***) in three-dimensional Cartesian coordinates*

**a,b,c***OXYZ*is

**normal**(perpendicular) to a plane

*described by an equation*

**α**

**a·x + b·y + c·z + d = 0**where

*,*

**a***,*

**b***,*

**c***are real numbers.*

**d***Proof*

CASE 1 (easy) - Constant

*in an equation that describes plane*

**d***equals to zero.*

**α**The equation for plane

*looks in this case as*

**α**

**a·x + b·y + c·z = 0**Then plane

*must go through the origin of coordinates*

**α***(*

**O***) because this point satisfies the equation for*

**0,0,0***.*

**α**Consider a vector from an origin of coordinate

*(*

**O***) to*

**0,0,0****any other point**on a plane

*(*

**Q***).*

**x,y,z**Obviously, non-null vector

*(*

**OQ***) is lying fully within plane*

**x,y,z***because both its ends - points*

**α***and*

**O***lie within this plane.*

**Q**We can interpret the equation

**a·x + b·y + c·z = 0**that describes plane

*in this case as a scalar product of non-null vector*

**α***(*

**n***) and non-null vector*

**a,b,c***(*

**OQ***).*

**x,y,z**Since this scalar product between vector

*(*

**n***) and any vector*

**a,b,c***(*

**OQ***) lying within plane*

**x,y,z***is zero, vector*

**α***(*

**n***) is perpendicular to plane*

**a,b,c***.*

**α**CASE 2 - Constant

*is not equal to zero.*

**d**Consider two planes defined by two equations

Plane

*:*

**α**

**a·x + b·y + c·z + d = 0**Plane

*:*

**β**

**a·x + b·y + c·z = 0**Since

*, any point that satisfies one of these equations will not satisfy another.*

**d ≠ 0**Therefore, these planes do not have any common points and, therefore, are parallel.

We have already proven that

**non-null**vector

*(*

**n***) is perpendicular to plane*

**a,b,c***(see CASE 1 above).*

**β**Consequently, this vector

*is perpendicular to*

**n***as well, that is*

**α***is*

**n****normal**to

*.*

**α**End of the proof that

**n****⊥**

*.*

**α**
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