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 xy-plane
 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 P1(x1, y1) and the direction vector s then, the position or (radius) vector r of any point P(x, y) of the line
r = r1 + t s,    - oo < t < + oo   and where,  r1 = x1i + y1 j  and  s = xsi + ys 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 = (x1 + xst) i + (y1 + yst) 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 = x1 + xs t
y = y1 + ys 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 P1(x1, y1) and P2(x2, y2)
x = x1 + (x2 - x1) t
y = y1 + (y2 - y1) 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, (x0, y0, z0) 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 = x0 + at,   y = y0 + bt  and  z = z0 + 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 + 4y = 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.
Pre-calculus contents C
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