Coordinate Geometry (Analytic Geometry) in Three-dimensional Space
     Points, lines and planes in three-dimensional coordinate system represented by vectors
      Equations of a line in space
      The vector equation of a line
         The parametric equations of a line
      Equation of a line defined by direction vector and a point - Symmetric equation of a line
      Line given by two points
         Distance between two given points
         Orthogonal projection of a line in space onto the xy coordinate plane
      A line in a 3D space examples
         Angle between lines
      Condition for intersection of two lines in a 3D space
Equations of a line in space
The vector equation of a line
Through a given point A(x0, y0, z0), which is determined by position vector r0 = x0i + y0 j + z0k, passes a line directed by its direction vector s = ai + bj + ck.
Thus, the position of any point P(x, y, z) of a line is then uniquely determined by a vector
   
which is called the vector equation of a line.
That is, a radius vector r = xi + y j + zk of every point of the line, represents the sum of the radius vector r0, of the given point, and a vector  t s collinear to the vector s, where t is a parameter which can take any real value from  - oo  to + oo .
 
The parametric equations of a line
By writing the above vector equation of a line in the component form
obtained are components of the vector r, x = x0 + at,   y = y0 + bt  and  z = z0 + ct  which, at the same
time, represent coordinates of any point of the line expressed as the function of a variable parameter t.
That is why they are called the parametric equation of a line.
Equation of a line defined by direction vector and a point - Symmetric equation of a line
Now, let express t from the above parametric equations
so, by equating obtained is
   
equation of a line passing through a point A(x0, y0, z0) and given direction vector s = ai + bj + ck.
Scalar components (or the coordinates), a, b, and c, of the direction vector s, are
   
or the direction cosines
 
That is, the cosines of the angles that a line forms by the coordinates axes x, y and z, or the scalar components of the unit vector of the direction vector s
   
Line given by two points
A line through points A and B, determined by their position vectors,
 r1 = x1i + y1 j + z1k  and  r2 = x2i + y2 j + z2k,
has the direction vector s = r2 - r1 so that its vector equation is
   
If we write this equation in the component form that is
by equating corresponding scalar components
and by eliminating parameter t
   
obtained is equation of a line through two given
pointsA(x1, y1, z1) and B(x2, y2, z2).
The direction cosines are,
where
is the distance between given points A and B.
If, for example, in the above equation of a line through two points in a space, we take that z coordinate of both given points is zero, we obtain known equation of a line through two points in a coordinate plane, i.e.,
and at the same time, it is the equation of the orthogonal projection of a line in 3D space onto the xy  coordinate plane.
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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