
Coordinate
Geometry (Analytic Geometry) in Threedimensional Space 

Plane
in a threedimensional (3D) coordinate system 
Comparison of
the general form and the
Hessian normal form of equations of a plane

The distance of a point to a plane 
plane given in general form

Angle (dihedral angle) between two
planes






Comparison of general form and the
Hessian normal form of equations of a plane

Coefficients,
A,
B
and C,
beside coordinates x,
y
and z, of the general form of equation 
Ax
+ By
+ Cz
+ D = 0 
are the components of the normal vector
N
= Ai + Bj
+ Ck, while in the Hessian’s equation 
x
· cosa
+ y · cosb
+ z · cosg

p
= 0, 
on that place, there are components (the direction cosines) of the unit vector
N°
of the same vector. 

Thus, to convert from general form to the Hessian normal form, divide the general form of equation by 

taking the sign of the square root opposite to the sign of
D, where
D
is not 0 



The distance of a point to a plane 
plane given in general form

Let
replace in the Hessian formula for the distance d
= x_{0}
· cosa
+ y_{0} · cosb
+ z_{0} · cosg 
p, the direction
cosines and the length of the normal p, by coefficients of the general form,
to obtain 

the formula for the
distance of a point to a plane given in the general
form. 

Angle (dihedral angle) between two
planes

The angle subtended by normals drawn from the origin to the first and second plane, that is, the angle which
form
the normal vectors of the given planes is 

the
angle between two planes. 
If planes are parallel normal vectors are collinear that is, 
N_{1}
= lN_{2}
<=> j
= 0°, 
and if planes are orthogonal, then their normal vectors are orthogonal too, so their scalar product is zero, 
N_{1} · N_{2}
= 0 <=> j
= 90°. 

Example:
Find the angle between planes,
P_{1}
::
2x 
y + 4z 
3 = 0 and
P_{2
}:: x 
3y + 5z + 6 = 0.

Solution:
Thus, 
N_{1} =
2i

j + 4k and
N_{2}
=
i

3j + 5k,
then 














Precalculus contents
J 



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