Coordinate Geometry (Analytic Geometry) in Three-dimensional Space
     Points, lines and planes in three-dimensional coordinate system represented by vectors
      A line in a 3D space examples
      Angle between lines
      Condition for intersection of two lines in a 3D space
A line in the 3D space examples
Example:   Determine equation of a line passing through the point A(-1, -2, 3) and which is parallel to the vector s = 2i + 4j + 2k.
Solution:
By plugging coordinates of the given point A and the components of the direction vector s into equation of a line
obtained is
 
Example:   Find the equation of a line passing through the points, A(1, 0, 2) and B(4, 5, 6).
Solution:
By plugging coordinates of the given points into equation of a line through two given points
obtained is
Plug the coordinates of both points into obtained equation to verify the result.
 
Example:   Find the angles that a line  forms with coordinate axes
Solution:  The unit vector of the direction vector s = -2i - j + 2k
Angle between lines
Angle between two lines  equals the
angle subtended by direction vectors, s1 and s2 of the lines
For the lines that do not intersects, i.e., for the skew lines (such as two lines not lying on the same plane in space), assumed is the angle between lines that are parallel to given lines that intersect.
That is, the initial points of their direction vectors always can be brought to the same point by translation.
Condition for intersection of two lines in a 3D space
Two lines in a 3D space can be parallel, can intersect or can be skew lines.
Two parallel or two intersecting lines lie on the same plane, i.e., their direction vectors, s1 and s2 are coplanar with the vector P1P2 = r2 - r1 drawn from the point P1, of the first line, to the point P2 of the second line.
Therefore, the scalar triple product of these vectors is zero,
   
Example:  Given are lines,  examine whether 
 lines intersect or are skew lines, and if intersect, find the intersection point and the angle between lines.
Solution:  
 In the given equations of lines,
         P1(1, -1, 4)  and  s1 = -3i + 4 j - 2k,
and 
       P2(3, 2, -2)  and  s2 = -5i + j + 4k
therefore, vector 
P1P2 = r2 - r1 = 2i + 3 j - 6k.
Let examine whether the lines intersect
 
therefore, the lines intersect.
Intersection of two lines is a point, coordinates of which satisfy both equations therefore, solutions, x, y and z of the equations, l1 and l2 are the coordinates of the intersection point, that is
Thus, given lines intersect at the point S(-2, 3, 2).
The angle between direction vectors, s1 and s2 of the lines,  we calculate from the formula
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