This document discusses vector and parametric equations that describe lines. It provides the following key points: 1) A vector equation of a line passing through a point P1 and parallel to a vector a is defined as P1P2 = t*a, where t is a scalar. 2) The parametric equations of a line parallel to a vector a and passing through a point P1 are x = x1 + t*a1 and y = y1 + t*a2, where t is any real number. 3) Each value of t in the parametric equations establishes a point (x,y) on the line. Setting the independent variable t equal in both equations allows

1. Precalculus
8-6 Vectors and Parametric Equations
Vector Equation of a Line
A line through a point P1 ( x1 , y1 ) which is _______________________ to the vector
r
a = α1 , α2 is defined by the _________ ________ _________________
r r
P1 ( x1 , y1 ) ανδ Π ( ξ2 , ψ ) such that the vector P1P2 is a ________________ multiple of a .
2 2
r ρ
Therefore, P1P2 = τα for some scalar t.
x2 − ξ1 , ψ − ψ = τ α1 , α2
2 2
EX1: Write a vector equation describing a line passing through P1 (1, 4 ) and parallel to
r
a = 3, −2 .
Parametric Equations of a Line
r
A line through P1 ( x1 , y1 ) that is parallel to a = α1 , α2 has the following
___________________ equations, where t is any real number (________________).
•
•
r
EX2: Find the parametric equations for a line parallel to q = 6,−3 and passing through
the point at ( −2,- 4 ) . Then make a table of values and graph the line.
t x y

2. Note that each value of t establishes an ________________________________________
( x, y ) that is a point on the line.
Parametric Equation ↔ Slope-Intercept Form
Given: Slope-Intercept Form
• x is the _________________________ variable, and y is the
________________________ variable.
• In parametric equations, t is the independent variable, and x and y are both
______________________ on t.
• If we set the independent variables x and t ______________, we can write _____
parametric equations in terms of t.
EX: Write parametric equations of y = −4 ξ + 7
Given: Parametric Equations
• _______________ both equations for t.
• Set the equations equal to _________________________________.
• Solve for y (put in slope-intercept form).
EX: Write an equation in slope-intercept form of the line whose parametric equations
are x = −5 + 4 τ ανδ ψ= 2 − 3τ .
HW p. 524 (12 – 30 even)