Download Fundamentals of Mathematics - §2.5

Fundamentals of Mathematics
§2.5 - Parallel and Perpendicular Lines
Ricky Ng
October 18, 2013
Ricky Ng
Fundamentals of Mathematics
Announcements
Quiz 8 is due today.
Ricky Ng
Fundamentals of Mathematics
Continue - What About Standard Form?
Write an equation in standard form for the line with slope =
and y-intercept 1.
Ricky Ng
§2.4 - Equations of Lines
1
3
Parallel Lines
Definition
Given two lines L1 and L2 with slopes m1 and m2 , respectively.
We say L1 is parallel to L2 if m1 = m2 .
For example, the lines
y = 2x − 1
and
y = 2x + 4
are parallel.
Ricky Ng
§2.5 - Parallel and Perpendicular Lines
Geometric Meaning
Ricky Ng
§2.5 - Parallel and Perpendicular Lines
Graphically speaking, two lines are parallel when they
“behave” the same: Travel in exactly the same direction.
Consequently,
Rule
If two lines are parallel, either they are the same line or
they do not intersect each other.
If two lines are not parallel, they intersect each other at
exactly one point.
Ricky Ng
§2.5 - Parallel and Perpendicular Lines
Example
Are the lines y = 34 x + 2 and 3x − 4y = 1 parallel?
Ricky Ng
§2.5 - Parallel and Perpendicular Lines
Example
Let L1 be the line that passes through the points (1, 4) and
(2, 2), L2 be the line that passes through the points (3, 4) and
(2, 6). Are L1 and L2 parallel lines?
Ricky Ng
§2.5 - Parallel and Perpendicular Lines
Popper 14: Question 1
Ricky Ng
§2.5 - Parallel and Perpendicular Lines
Example
Find an equation for the line that passes through (−2, 1) and is
parallel to the line y = −3x + 16.
Ricky Ng
§2.5 - Parallel and Perpendicular Lines
Example
Find the equation in slope-intercept form for the line with
y-intercept 2 that is parallel to the line 2x − y = 3.
Ricky Ng
§2.5 - Parallel and Perpendicular Lines
Popper 14: Question 2
Ricky Ng
§2.5 - Parallel and Perpendicular Lines
Perpendicular Lines
Definition
Given two lines L1 and L2 with slopes m1 and m2 , respectively.
We say L1 is perpendicular to L2 if
m1 × m2 = −1,
or equivalently,
m1 = −
1
.
m2
For example, the lines
1
y = x+2
2
are perpendicular because
and
1
2
y = −2x + 4
× (−2) = −1.
Ricky Ng
§2.5 - Parallel and Perpendicular Lines
Geometric Meaning
Ricky Ng
§2.5 - Parallel and Perpendicular Lines
Graphically speaking, L1 and L2 are perpendicular when they
intersect at a right angle.
Therefore...
Remark
If L1 is a horizontal line (m1 = 0) and L2 is vertical line
(m2 is undefined), we still say L1 is perpendicular to L2 ,
although m1 × m2 cannot be computed.
Ricky Ng
§2.5 - Parallel and Perpendicular Lines
Example
Are the lines y = 23 x + 2 and y − 3 = − 23 (x + 2) perpendicular?
Ricky Ng
§2.5 - Parallel and Perpendicular Lines
Example
Let L be the line passing through (2, 4) and (7, 3). Is L
perpendicular to the line 10x − 2y = 3 ? Parallel? Or neither?
Ricky Ng
§2.5 - Parallel and Perpendicular Lines
Popper 14: Question 3
Ricky Ng
§2.5 - Parallel and Perpendicular Lines
Example
Give the equation in slope-intercept form for the line with
y-intercept 3 that is perpendicular to the line y = − 34 x + 2.
Ricky Ng
§2.5 - Parallel and Perpendicular Lines
Example
Give the equation in slope-intercept form for the line passing
through (1, 3) that is perpendicular to the line y = 2x − 1.
Ricky Ng
§2.5 - Parallel and Perpendicular Lines
Popper 14: Question 4
Ricky Ng
§2.5 - Parallel and Perpendicular Lines