1. 8-6: Vectors and Parametric Equations

2. Objectives Write vector and parametric equations of lines.
Graph parametric equations.

3. Applications
For objects moving in a straight line, there is a more useful way of writing equations for the line describing the object's path using vectors.

4. Vector Equations

5. Vector Equation of a Line
A line through P1(x1,y1) parallel to vector
is defined by the set of points P1(x1,y1) and P2(x2,y2) such that
for some real number t. Therefore,

6. Example
Write a vector equation describing a line passing through P1(-9,3) and parallel to
An equation like above can be used to describe the coordinates for a point on the line for any value of t (often time).

7. Parametric Equations of a Line
A line through P1(x1,y1) that is parallel to vector
has the following parametric equations where t is any real number.

8. Example
Find the parametric equations for a line parallel to and passing through the point at (4,3). Then make a table of values and graph.
Evaluating the parametric equations for a value of t gives the coordinates of the position of the object after t units of time. t x y -1 1 8 4 3 7 -2

9. Example Write the parametric equations of y=1/2 x + 7.
(parametric t: independent and x&y: dependent)
(in given equation: x is independent and y is dependent)
t=x y=1/2 t + 7
If you make a table of values and graph these parametric equations, you'll see that it's the same graph as y = 1/2 x + 7.
parametric equations

10. Example
Write an equation in slope-intercept form of the line whose parametric equations are
x= -2 + t and y=4-3t.
(Solve both equations for t and set them equal to each other.)

11. Homework
8-6: p. 524
# mult of 3, 37, 11