Math 151
Section 1.3
Vector Functions and Parametric Curves
Parametric Curve Let x be functions of t, where t is a parameter. As t varies over its domain, the
collection of points (x, y) = (x(t), y(t)) defines a parametric curve.
Example: Sketch the given parametric curves.
x (t ) = t ! 3
A.
y (t ) = 2t !1
B.
C.
x (t ) = 1! 2t
y (t ) = 2 + 3t
x (t ) = t +1
y (t ) = t 2 ! 4
for ! 3 " t < 3
Math 151
D.
E.
x (t ) = t
y (t ) = 1! t
x (t ) = 2sin !
y (t ) = 3cos!
Math 151
Vector Functions For each value of t, consider the point (x, y) = (x(t), y(t)) on the parametric curve
to be the terminal point of a vector r (t ) = x (t ), y (t ) originating from the origin. r defines a vector
function.
Example: Sketch the given curves as defined by the vector functions. Include the direction of the
curve as t increases.
A. r (t ) = t ! 3,2t !1
B. r (t ) = 2 + cost,1+ sin t
Math 151
Vector Equation of a Line The vector equation of a line passing through the point r0 = ( x0 , y0 )
and parallel to the vector v = v1 ,v2 is given by r (t ) = r0 + tv .
The corresponding parametric equations of the line are given by
x (t ) = x0 + tv1
y (t ) = y0 + tv2
Example: Find a vector function of the line parallel to the vector 1,4 and passing through the
point (−1, 5).
Example: Find parametric equations for the line with slope
4
and passing through the point (2, −5).
3
Math 151
Example: An object is moving in the xy-plane and its position after t seconds is given by
r (t ) = t + 4,t 2 + 2 .
A. What is the position of the object at time t = 2?
B. At what time does the object reach the point (7, 11)?
C. Does the object pass through the point (9, 20)?
D. Eliminate the parameter to find the Cartesian equation.