Section 1.3: Parametric Curves and Vector Functions
Definition: The equations x = x(t) and y = y(t) are called parametric equations with
parameter t. As t varies over its domain, the points (x, y) = (x(t), y(t)) trace out a
parametric curve.
Note: To graph a parametric curve, find a relationship between x and y to eliminate the
parameter t and obtain a Cartesian equation of x and y.
Example: Sketch the curve described by the following parametric equations.
(a) x(t) = t − 2, y(t) = 2t − 1
(b) x(t) = 5t − 3, y(t) = t2 − 4
(c) x(t) = 3 sin θ, y(t) = 2 cos θ, 0 ≤ θ ≤ 2π
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~
Definition: A vector function in R2 is a function R(t)
= hx(t), y(t)i that assigns a unique
2
vector in R to every value of t in its domain. The functions x(t) and y(t) are called the
~
component functions of R.
~
Note: The domain of a vector function is the set of all t ∈ R such that R(t)
is defined. That
is, the set of all real values of t for which x(t) and y(t) are defined.
Example: Sketch the curve described by the following vector functions. Indicate the direction
of the curve for increasing t.
~
(a) R(t)
= ht + 3, 2 − 4ti
~
(b) R(t)
= h2 + cos t, 3 + sin ti, 0 ≤ t ≤ 2π
π
~
(c) R(t)
= hsin t, csc ti, 0 < t ≤
2
2
Example: An object is moving in the xy-plane and its position after t seconds is given by
~
R(t)
= ht − 1, t2 − 2ti.
(a) Find the position of the object at time t = 6.
(b) At what time does the object reach the point (3, 8)?
(c) Does the object pass through the point (6, 39)?
(d) Find a Cartesian equation for the path of the object.
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Definition: The line containing the point (x0 , y0 ) and parallel to the vector ~v = hA, Bi has
parametric equations
x(t) = x0 + At
y(t) = y0 + Bt,
where t ∈ R is the parameter. These equations can be expressed as a vector function
~
R(t)
= hx0 + At, y0 + Bti.
Example: Find parametric equations of the line passing through (−3, 7) and parallel to the
vector ~v = h5, −9i.
Example: Find a vector equation for the line passing through the points (−3, 4) and (2, 8).
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~ 1 (t) = h−4+2t, 5+ti
Example: Determine whether the lines defined by the vector functions R
~ 2 (s) = h2+3s, 4−6si are parallel, perpendicular, or neither. If the lines are not parallel,
and R
find their point of intersection.
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