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
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.
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