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| Coordinate
Geometry (Analytic Geometry) in Three-dimensional Space |
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Points, lines and planes in three-dimensional coordinate
system represented by vectors |
Equations of a line in space |
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The
vector equation of a line
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The parametric equations of a line
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Equation of a line defined by
direction vector and a point - Symmetric equation of a line |
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Line given by two points
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Distance between two given points
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Orthogonal projection of a line in
space onto the xy
coordinate plane |
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A
line in a 3D space examples |
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Angle between lines
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Condition for intersection of two
lines in a 3D space
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| Equations of a line in space |
| The
vector equation of a line
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| 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 |
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| 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
. |
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| The parametric equations of a line
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| By writing the above vector equation of a line in
the component form
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| obtained are components of the vector
r, |
x
= x0 + at,
y = y0 +
bt and z
= z0 + ct |
which, at the same
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| time, represent coordinates of any point of the line expressed as the function of a variable
parameter
t.
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| That is why they are called the
parametric equation of a line.
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| 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
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| so, by equating
obtained is
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| 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 |
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| or the direction cosines
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| That
is, the cosines of the angles that a line forms by the coordinates axes
x,
y
and
z, or the scalar
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| components of
the unit vector of the direction vector
s
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| Line given by two points
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| 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 |
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| If we write this equation in the component form that is |
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| by equating corresponding scalar components |
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| and by
eliminating parameter t |
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| obtained is
equation of a line through two given |
| points, A(x1,
y1, z1)
and B(x2,
y2, z2). |
| The direction cosines are, |
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| where |
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| 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
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| given points
is zero, we obtain known equation of a line through two points in a coordinate plane, i.e., |
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| and at the same time,
it is the equation of the orthogonal projection of a line
in 3D space onto the xy
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| coordinate
plane.
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| Example:
Determine equation of a line passing through the point
A(-1,
-2, 3) and which is parallel to the
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| vector
s
= 2i + 4j + 2k. |
| Solution: |
| By plugging coordinates of the given point A
and |
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the components of the direction vector s
into |
| equation of a line |
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| obtained is |
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| Example:
Find the equation of a line passing through the points,
A(1,
0, 2) and B(4,
5, 6).
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| Solution: |
| By plugging coordinates of the given points into |
| equation of a line
through two given points |
 |
| obtained is |
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| Plug the coordinates of both points into obtained equation to verify
the result.
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| Example:
Find the angles that a line
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forms with coordinate
axes
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| Solution: The unit vector of the direction vector
s
= -2i
-
j + 2k |
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| Angle between lines
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| Angle between two lines |
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equals the |
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| angle
subtended by direction vectors, s1
and s2
of the lines |
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| 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. |
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| Condition for intersection of two
lines in a 3D space
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| 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
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| 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, |
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| Example: Given are lines, |
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examine whether |
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| lines intersect or are skew lines, and if intersect, find the intersection point and the angle between lines. |
| Solution:
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| In the given equations of lines, |
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P1(1, -1,
4) and
s1
= -3i + 4
j -
2k,
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| and |
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P2(3, 2, -2)
and s2
= -5i +
j + 4k |
| therefore,
vector |
| P1P2
= r2 -
r1
= 2i + 3 j -
6k. |
| Let
examine whether the lines intersect |
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 |
| 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 |
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| 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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| Coordinate
geometry contents |
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