
Coordinate
Geometry (Analytic Geometry) in Threedimensional Space 


Points, lines and planes in threedimensional 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(x_{0},
y_{0}, z_{0}), which is determined by position vector
r_{0}
= x_{0}i + y_{0 }j
+ z_{0}k, 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 r_{0},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
= x_{0} + at,
y = y_{0} +
bt and z
= z_{0} + 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(x_{0},
y_{0}, z_{0})
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, 
r_{1}
= x_{1}i + y_{1 }j
+ z_{1}k
and r_{2}
= x_{2}i + y_{2 }j
+ z_{2}k, 
has the direction vector
s
= r_{2} 
r_{1} 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 
points, A(x_{1},
y_{1}, z_{1})
and B(x_{2},
y_{2}, z_{2}). 
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.










Precalculus contents
J 



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