

Parametric Equations 
Parametric
equations definition 
Use
of parametric equations, example 





Parametric
equations definition 
When
Cartesian coordinates of a curve or a surface are represented as
functions of the same variable (usually written t),
they are
called the parametric equations. 
Thus,
parametric equations in the xyplane 
x
= x (t)
and y
= y (t) 
denote
the x
and y
coordinate of the graph of a curve in the plane. 

The parametric equations of a line

If
in a coordinate plane a line is defined by the point P_{1}(x_{1},
y_{1}) and the
direction vector s
then, the position or (radius)
vector r
of any point P(x,
y) of the line 
r
= r_{1} + t · s,

oo
< t < + oo
and
where, r_{1}
= x_{1}i + y_{1} j
and s
= x_{s}i + y_{s} j, 
represents the vector equation of the line. 
Therefore,
any point of the line can be reached by the 
radius vector 
r
= xi + y j
= (x_{1} + x_{s}t) i
+ (y_{1} + y_{s}t) j 
since the scalar quantity t
(called the
parameter) can take any real value from 
oo
to + oo. 
By
writing the scalar components of the above vector 
equation obtained is 
x
= x_{1} + x_{s} · t 
y
= y_{1} + y_{s} · t 


the parametric
equations of the line. 



To
convert the parametric equations into the Cartesian coordinates solve
given equations for t.
So 

by
equating 


Therefore,
the parametric equations of a line passing through
two points P_{1}(x_{1},
y_{1}) and P_{2}(x_{2},
y_{2}) 
x
= x_{1} + (x_{2} 
x_{1}) t 
y
= y_{1} + (y_{2} 
y_{1}) t 



Use
of parametric equations, example 
Intersection point of a line and a plane
in three dimensional space

The point of intersection is a common point of a line and a plane. Therefore, coordinates of intersection must
satisfy both equations, of the line l
and the plane P 


and 



where, (x_{0},
y_{0},
z_{0}) is a
given point of the line and s =
ai + bj + ck
is direction vector of the line, and 
N =
Ai + Bj + Ck
is the normal vector of the given plane. 
Let
transform equation of the line into the parametric form 

Then,
the parametric equation of a line, 
x =
x_{0} + at,
y = y_{0}
+ bt and z
= z_{0} + ct 
represents coordinates of any point of
the line expressed as the function of a variable parameter t
which makes
possible to determine any point of the line according to a given condition. 
Therefore, by plugging these variable coordinates of a point of the line into
the given plane determine what value
must have the parameter t,
this point to be the common point of the line and the plane. 

Example:
Given is a line 

and a plane
4x 
13y + 23z 
45 = 0, find the


intersection
point of the line and the plane. 
Solution: Transition from the symmetric to the parametric form
of the line


by plugging these variable coordinates
into the given plane we will find the value of the parameter t
such that these
coordinates represent common point of the line and the plane, thus 
x =
t +
4, y
= 4t 
3 and z
= 4t 
2 =>
4x 
13y + 23z 
45 = 0 
which
gives, 4 · (t +
4)  13 · (4t 
3) + 23 · (4t 
2) 
45 = 0 => t = 1. 
Thus,
for t = 1
the point belongs to the line and the plane, so 
x =
t +
4 = 
1+ 4 = 3, y =
4t 
3 = 4 · 1 
3 = 1 and z =
4t 
2 = 4 · 1 
2 = 2. 
Therefore, the intersection point
A(3,
1,
2) is the point which
belong to both, the line and the plane, prove. 








Precalculus
contents C 



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