on a line parallel to the unit vecotor 
, which passes point 
, is represented as follows, using parameter 
.Equation of line in 2D plane and 3D space is shown. Formula of plane is also shown. The distance between point and line is shown. Perpendicular foot from a point to a line is shown. Distance between point and plane is shown. Perpendicular foot from a point to a plane is shown. Mathematical representation of the intersection between two lines in both 2D coordinates and 3D coordinates is also shown.
Point on a line parallel to the unit vecotor , which passes point , is represented as follows, using parameter .
Denote 
, 
, 
, and reformulate, we obtain the following.
As for 2D, 
, namely 
.
Point on a line passing point 
and point 
is represented as follows, using parameter .
Denote 
, 
, and reformulate, we obtain the following.
As for 2D, 
, namely 
.
Point on a plane including point , whose unit normal vector is 
, is represented as follows.
Denote , 
, , and reformulate, we obtain the following.
Intersection point between a line and a perpendicular line from point is represented as follows. Note that must be a unit vector.
Thus, using the above , the distance between point and line is represented as follows.
Let's think about a line parallel to a unit vector 
and passes point 
. Defining a unit vector 
orthogonal to vector , the distance between point and line is represented as follows.
By the way, the line is or 
.
Let's redefine as 
, 
, 
. The line equation becomes following equation.
Representing the distance between point and line using 
, 
, 
results in the following formula.
To summarize, the distance between point 
and line , when 
holds, is the following.
Intersecting point between a plane and a perpendicular line from point is represented as follows. must be unit vector.
The distance between point and plane is represented as follows.
By the way, plane equation is or 
.
Let's redefine as 
,
, 
, 
. The plane equation becomes following equation.
As a result, the distance between point and plane , when 
holds, is represented as follows.
Intersection point between line 
and line 
is as follows.
Let's think about a intersecting point between 3D line 
and 3D line 
. 
and 
are unit vectors. Generally, 2 lines in 3D have no intersecting point.
Let's calculate a point which is the closest to 2 lines.
Point should be close to point , so 
. Point should be close to point , so 
. Therefore, the following is derived.
Therefore,
Or,
Multiply 
from the left for both sides of this equation.
Therefore, the following holds.
Solving this results in the following.
The solution which satisfies both and as much as possible is 
. Substituting the above equation to this equation results in the following
intersecting point.
Here, 
holds.
If 
, intersecting point exists.
If 
, intersecting point does not exist because 2 lines are parallel.