
Coordinate
Geometry (Analytic Geometry) in Threedimensional Space 

Points, lines and planes in threedimensional 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, 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 









Precalculus contents
J 



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