Name _________________
Worksheet 8.7
Parametric Equations
0. Introduction
By working through this worksheet you will learn how to specify a line in three dimensions.
Doing so will be a little harder than doing so in two dimensions. In 2-D, a typical x-y equation,
such as 4 x  3 y  7 , specifies a line. It would thus seem reasonable to suspect that something
like 4 x  3 y  5 z  7 would specify a line in 3-D, but sadly it does not. Instead such an equation
represents an entire plane!
1. Parametric Equations
Representing a line using parametric equations is often more practical than typical x-y equations,
and as we will soon see, can be extended to higher dimensions more easily. Consider this simple
pair of parametric equations.
3
x  2 t
5
4
y  3 t
5
Here t is considered the independent parameter, while x and y are both dependent variables.
(a) Sketch the points (x, y) when t = −5, 0, 5, 10, 15
(b) What does the set (often called the “locus”) of points (x, y) look like as t takes all possible
values?
(c) Give the equation of this locus in functional form y = f(x). This will be easiest if you find t as
a function of y first.
(d) Convert this to ax  by  c form.
(e) Describe the connections between this last equation and the two original vector parametric
equations.
4
(f) What is the relationship between this line and the vector   ?
3
2. Two Dimensional Line
Consider the points (1, 2) and (3,1)
(a) Find the vector u that points from the origin to (1, 2)
(b) Find vector v that points from (1, 2) to (3,1)
(c) Find the unit vector v̂ that points in the same direction as v
(d) Create an expression for any point on the line by adding a scalar multiple (use the variable t
to denote the scalar) of v̂ to the position vector u of (1, 2)
(e) The (parametric) vector equation (in the form x  u  t  v ) that you just created should give
you a coordinate for any given value of t just like a system of parametric equations. Find the
coordinate when t = 3.
(f) How far away is this point from a point obtained by setting t = −2 ?
(g) Draw a diagram to represent everything that you have done so far.
(h) Now change the vector equation to a pair of parametric equations.
(i) Now eliminate t and put this line into ax  by  c
(j) Verify that your original two points (1, 2) and (3,1) satisfy this equation.
3. Two Dimensional Practice
(a) Find a parametric vector equation of the line through (−1, 3) and parallel to 3x − 4y = 7
Include a sketch that represents the vector equations
(b) Find a parametric vector equation of the line perpendicular to 3x – 4y = 7. Ensure that x = 2
and y = 3 when t = 0. Again, include a sketch
4. Three Dimensional Line
(a) Find a vector equation of the line through the points (3, −2, 1) and (4, 0, 5)
(b) Find a vector equation of the line through the origin and perpendicular to the plane
3x  2 y  5 z  0
Bonus. Lowering the Dimension
(a) Find a way to specify any single point in two dimensions using only one equal sign (one
equation). If you succeed, then please relate your discovery immediately to the nearest
mathematically savvy friend. If you fail, think about how this relates to the problem of
specifying a line in three dimensions using only one equation.
(b) In one dimension (also known as the real number line) how many equations do you need to
specify a single point?
(c) Recalling that a point is zero dimensions, that a line is one dimension, and that a plane is two
dimensions, come up with a general rule about what one equation allows you to specify in ndimensional space.
(d) Use your rule to predict the minimum number of equations needed to specify a line in three
dimensions?