
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 
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(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.



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, s_{1}
and s_{2}
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, s_{1}
and s_{2} are coplanar with the vector
P_{1}P_{2}
= r_{2} 
r_{1} drawn
from the point P_{1},
of the first line, to the point
P_{2 }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, 
P_{1}(1, 1,
4) and
s_{1}
= 3i + 4
j 
2k,

and 
P_{2}(3, 2, 2)
and s_{2}
= 5i +
j + 4k 
therefore,
vector 
P_{1}P_{2}
= r_{2} 
r_{1}
= 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,
l_{1}
and l_{2}
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, s_{1}
and s_{2}
of the lines, we calculate from the formula 










Coordinate
geometry contents 



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