7.6 NORMAL FORM OF A LINEAR
EQUATION
By the end of the section students will be able to
write the standard form of a linear equation given
the length of the normal and the angle it makes with
the x-axis and write linear equations in standard
form as evidenced by a mix-match activity.
By the end of the section students will be able to write the standard form of a linear equation given the length of the normal
and the angle it makes with the x-axis and write linear equations in standard form as evidenced by a mix-match activity.
WHAT IS A “NORMAL” LINE
We find an equation of a line using a point and a
slope
 Distance between two points comes from the
Pythagorean theorem


𝑎2 + 𝑏2 = 𝑐 2 → 𝑐 = ± 𝑎2 + 𝑏2
sin 𝜃
cos 𝜃

Slope can be found using

“Normal” mean perpendicular
=
𝑟𝑖𝑠𝑒
𝑟𝑢𝑛

Perpendicular slopes are OPPOSITE sign and
RECIPROCAL

−
cos 𝜃
sin 𝜃
By the end of the section students will be able to write the standard form of a linear equation given the length of the normal
and the angle it makes with the x-axis and write linear equations in standard form as evidenced by a mix-match activity.
SIDES OF A TRIANGLE

𝜙 - Greek letter can be pronounced either
fee (as in a bank fee) or
 fi (rhymes with pie)


sin 𝜙 =



→ 𝑦 = 𝑝 ∙ sin 𝜙
cos 𝜙 =

𝑦
𝑝
𝑥
𝑝
𝑝
𝑦
𝜙
𝑥
→ 𝑥 = 𝑝 ∙ cos 𝜙
Thus, the sides of our triangle can be found using
the angle and the length of the hypotenuse
By the end of the section students will be able to write the standard form of a linear equation given the length of the normal
and the angle it makes with the x-axis and write linear equations in standard form as evidenced by a mix-match activity.
WHAT IS A “NORMAL” LINE

𝑝 is the distance between
the origin and the line


y
B
𝜙
𝜙
(distance is measured
PERPENDICULARLY)
C
𝜙 is the angle made with the
positive x-axis
p
A
𝜙
O
x
M
We want
the equation
of THIS line
By the end of the section students will be able to write the standard form of a linear equation given the length of the normal
and the angle it makes with the x-axis and write linear equations in standard form as evidenced by a mix-match activity.
WHAT IS A “NORMAL” LINE

y
cos 𝜙
sin 𝜙
𝑚=−
How do we get the normal line?
𝑦 − 𝑦1 = 𝑚 𝑥 − 𝑥1
B
𝜙
𝜙
𝑝 ∙ cos 𝜙 , 𝑝 ∙ sin 𝜙
𝑥1
cos 𝜙
𝑦 − 𝑝 sin 𝜙 = −
(𝑥 − 𝑝 cos 𝜙)
sin 𝜙
sin 𝜙
𝑚=
cos 𝜙
C
p
𝑝 ∙ sin 𝜙
A
𝜙
𝑦 sin 𝜙 − 𝑝 sin2 𝜙 = − cos 𝜙 (𝑥 − 𝑝 cos 𝜙)
O
𝑦1
x
M
𝑦 sin 𝜙 − 𝑝 sin2 𝜙 = −𝑥 cos 𝜙 + 𝑝 cos 2 𝜙
𝑥 cos 𝜙 + 𝑦 sin 𝜙 − 𝑝 sin2 𝜙 − 𝑝 cos2 𝜙 = 0
𝑥 cos 𝜙 + 𝑦 sin 𝜙 − 𝑝 sin2 𝜙 + cos2 𝜙 = 0
𝑥 cos 𝜙 + 𝑦 sin 𝜙 − 𝑝 = 0
𝑝 ∙ cos 𝜙
We want
the equation
of THIS line
By the end of the section students will be able to write the standard form of a linear equation given the length of the normal
and the angle it makes with the x-axis and write linear equations in standard form as evidenced by a mix-match activity.
HOW IS NORMAL FORM DIFFERENT THAN
STANDARD FORM?
Normal Ratio coefficients



−
1
𝑥
2
2
𝑥
2
−
+
3
𝑦
2
2
𝑦
2
−3 2=0
Positive leading coeff.
Standard Non ratio coefficients

𝑥−𝑦+6=0

𝑥 − 3𝑦 − 30 = 0

−
− 15 = 0
3𝑥 − 𝑦 + 10 = 0
3
𝑥
2
1
2
+ 𝑦−5=0
Why do you have a problem with the last bullet points?
What is the difference between Normal and Standard?
By the end of the section students will be able to write the standard form of a linear equation given the length of the
normal and the angle it makes with the x-axis and write linear equations in standard form as evidenced by a mix-match
activity.
EXAMPLE 1: WRITE THE NORMAL FORM OF THE EQUATION GIVEN BY
THE LENGTH OF THE NORMAL SEGMENT AND THE ANGLE MADE WITH
THE POSITIVE X AXIS
A.
𝑝 = 6, 𝜙 = 150°
𝑥 cos 𝜙 + 𝑦 sin 𝜙 − 𝑝 = 0
y
A

𝑥 cos 𝜙 + 𝑦 sin 𝜙 − 𝑝 = 0
𝑥 cos 150° + 𝑦 sin 150° − 6 = 0
3
1
−
𝑥+ 𝑦−6=0
2
2

x




𝑝 = 8, 𝜙 = 45°
𝑥 cos 45° + 𝑦 sin 45° − 8 = 0
2
2
𝑥+
𝑦−8=0
2
2



B
B.

y


x








By the end of the section students will be able to
to write
writethe
thestandard
standardform
formofofa a
linear
linear
equation
equation
given
given
the length
the length
of the of
normal
the
normal
and
the angle
and the
it makes
anglewith
it makes
the x-axis
withand
thewrite
x-axis
linear
andequations
write linear
in standard
equationsform
in standard
as evidenced
form as
by evidenced
a mix-match
by a
activity.
mix-match
activity.
EXAMPLE 1: WRITE THE NORMAL FORM OF THE EQUATION GIVEN BY
THE LENGTH OF THE NORMAL SEGMENT AND THE ANGLE MADE WITH
THE POSITIVE X AXIS
C.
𝑝 = 10 2, 𝜙 = 225
𝑥 cos 𝜙 + 𝑦 sin 𝜙 − 𝑝 = 0
y
C

𝑥 cos 225° + 𝑦 sin 225° − 10 2 = 0
2
2
−
𝑥+−
𝑦 − 10 2 = 0
2
2

x







D.
𝑝 = 4, 𝜙 = 300°
𝑥 cos 300° + 𝑦 sin 300° − 4 = 0
1
− 3
𝑥+
𝑦−4=0
2
2

D
y


x








By the end of the section students will be able to write the standard form of a linear equation given the length of the normal
and the angle it makes with the x-axis and write linear equations in standard form as evidenced by a mix-match activity.
HOW DO WE CONVERT FROM STANDARD
FORM TO NORMAL FORM?
Normal

2
𝑥
2
−
Standard
2
𝑦
2
+
−3 2=0




1
𝑥
2
−
−
3
𝑥
2
3
𝑦
2
− 15 = 0
1
2
+ 𝑦−5=0

𝑥−𝑦+6=0
Divide
everything by
𝑥 − 3𝑦 − 30 = 0
± 𝑨𝟐 + 𝑩𝟐 ,
use the
+ 3𝑥 −1𝑦 +10= 0 opposite sign of
the value for C
𝐵
𝐴
𝐶
𝐴2 + 𝐵 2
𝑝
𝑦
𝜙
𝑥
𝐵
𝜙
𝐴
By the end of the section students will be able to write the standard form of a linear equation given the length of the normal
and the angle it makes with the x-axis and write linear equations in standard form as evidenced by a mix-match activity.
WHERE ARE ALL THE ANGLES??

Which quadrant is being described by each?
sin 𝜙 < 0, cos 𝜙
 sin 𝜙 < 0, cos 𝜙
 sin 𝜙 > 0, cos 𝜙
 sin 𝜙 > 0, cos 𝜙


<0
>0
>0
<0
What is the measure of an angle?
1
2
3
Values like 0, ± , ± , ± , ±1 are found on the unit
2
2
2
circle, we can give EXACT angles
 Ratios that we don’t recognize can still be found using
𝑦
−1
𝑡𝑎𝑛
≈

𝑥
By the end of the section students will be able to write
writethe
thestandard
standard
form
form
of aof
linear
a linear
equation
equation
given given
the length
the length
of the of
normal
the
and the and
normal
angle
the
it angle
makesitwith
makes
thewith
x-axis
theand
x-axis
write
and
linear
writeequations
linear equations
in standard
in standard
form as form
evidenced
as evidenced
by a mix-match
by a mix-match
activity.
activity.
EXAMPLE 2: WRITE THE STANDARD FORM OF THE EQUATION
AND IDENTIFY 𝑝 AND 𝜙 GIVEN THE NORMAL FORM
𝐴𝑥 + 𝐵𝑦 + 𝐶 = 0
A.
6𝑥 + 8𝑦 + 3 = 0
What do we divide by?
± 𝐴2 + 𝐵2
± 62 + 82 = ±10
How do we know which sign to use?
opposite of C
+3 → −
6
8
3
− 𝑥− 𝑦−
=0
10
10
10
c𝑜𝑠 𝜙 < 0, sin 𝜙 < 0
3
4
3
− 𝑥− 𝑦−
=0
What quadrant is
5
5
10
this angle in?
𝐴, 𝐵, 𝐶 are not fractions
𝐴>0
𝑥 cos 𝜙 + 𝑦 sin 𝜙 − 𝑝 = 0
𝑝
4
tan 𝜙 =
3
𝜙 ≈ 𝑡𝑎𝑛−1
4
3
For the HW this
is acceptable,
UNLESS it’s a
unit circle value
𝜙 ≈ 53°
??? WHAT??? This is not in quadrant 3??
The calculator gives you the PRINCIPAL value, you need to translate that to
the appropriate quadrant
3
𝜙 = 233°
𝑝=
,
10
𝜙 = 233°
By the end of the section students will be able to write
writethe
thestandard
standard
form
form
of aof
linear
a linear
equation
equation
given given
the length
the length
of the of
normal
the
and the and
normal
angle
the
it angle
makesitwith
makes
thewith
x-axis
theand
x-axis
write
and
linear
writeequations
linear equations
in standard
in standard
form as form
evidenced
as evidenced
by a mix-match
by a mix-match
activity.
activity.
EXAMPLE 2: WRITE THE STANDARD FORM OF THE EQUATION
AND IDENTIFY 𝑝 AND 𝜙 GIVEN THE NORMAL FORM
𝐴𝑥 + 𝐵𝑦 + 𝐶 = 0
B.
3𝑥 − 1𝑦 − 8 3 = 0
What do we divide by?
± 𝐴2 + 𝐵2
𝐴, 𝐵, 𝐶 are not fractions
𝐴>0
2
±
3 + −1 2 = ±2
How do we know which sign to use?
opposite of C
−8 3 → +
3
1
8 3
𝑥− 𝑦−
=0
2
2
2
cos 𝜙 > 0, cos 𝜙 < 0
3
1
What quadrant is
𝑥− 𝑦−4 3=0
2
2
this angle in?
𝑝
−1
tan 𝜙 =
3
− 3
𝜙 = 𝑡𝑎𝑛−1
3
𝜙 = 150°, 330°
Which of these will give use the quadrant we want?
𝜙 = 330°
𝑥 cos 𝜙 + 𝑦 sin 𝜙 − 𝑝 = 0
For the HW this
is acceptable,
UNLESS it’s a
unit circle value
𝑝 = 4 3,
𝜙 = 330°
By the end of the section students will be able to write
writethe
thestandard
standard
form
form
of aof
linear
a linear
equation
equation
given given
the length
the length
of the of
normal
the
and the and
normal
angle
the
it angle
makesitwith
makes
thewith
x-axis
theand
x-axis
write
and
linear
writeequations
linear equations
in standard
in standard
form as form
evidenced
as evidenced
by a mix-match
by a mix-match
activity.
activity.
EXAMPLE 2: WRITE THE STANDARD FORM OF THE EQUATION
AND IDENTIFY 𝑝 AND 𝜙 GIVEN THE NORMAL FORM
𝐴𝑥 + 𝐵𝑦 + 𝐶 = 0
C.
5𝑥 − 12𝑦 + 65 = 0
𝐴, 𝐵, 𝐶 are not fractions
𝐴>0
What do we divide by?
± 𝐴2 + 𝐵2
± 52 + −12 2 = ±13
How do we know which sign to use?
opposite of C
+65 → −
5
12
65
− 𝑥+ 𝑦−
=0
sin 𝜙 < 0, cos 𝜙 > 0
13
13
13
5
12
𝑥 cos 𝜙 + 𝑦 sin 𝜙 − 𝑝 = 0
What quadrant is
− 𝑥+ 𝑦−5=0
this angle in?
13
13
𝑝
−12
tan 𝜙 =
5
For the HW this
−12
is acceptable,
𝜙 ≈ 𝑡𝑎𝑛−1
UNLESS
it’s a
5
unit circle value
𝜙 ≈ −67°
This angle is in quadrant 4 so
𝑝 = 5,
𝜙 = −67° = 293°
𝜙 = 113°
What if our angle was in quadrant 2?
Use 67° as the reference angle for QII → 180° − 67° = 113°
By the end of the section students will be able to write the standard form of a linear equation given the length of the
normal and the angle it makes with the x-axis and write linear equations in standard form as evidenced by a mix-match
activity.
EXAMPLE 2: WRITE THE STANDARD FORM OF THE EQUATION
AND IDENTIFY 𝑝 AND 𝜙 GIVEN THE NORMAL FORM
𝐴𝑥 + 𝐵𝑦 + 𝐶 = 0
D.
1𝑥 − 1𝑦 − 10 = 0
What do we divide by?
𝐴, 𝐵, 𝐶 are not fractions
𝐴>0
± 𝐴2 + 𝐵2
± 12 + −1
2
=± 2
10
10 2
=
=5 2
2
2
𝑝 = 5 2,
𝜙 = 315°
By the end of the section students will be able to write the standard form of a linear equation given the length of the
normal and the angle it makes with the x-axis and write linear equations in standard form as evidenced by a mix-match
activity.
EXAMPLE 2: WRITE THE STANDARD FORM OF THE EQUATION
AND IDENTIFY 𝑝 AND 𝜙 GIVEN THE NORMAL FORM
𝐴𝑥 + 𝐵𝑦 + 𝐶 = 0
D.
1𝑥 − 1𝑦 − 10 = 0
What do we divide by?
± 𝐴2 + 𝐵2
± 12 + −1 2 = ± 2
How do we know which sign to use?
opposite of C
−10 →+
1
1
10
𝑥−
𝑦−
=0
2
2
2
sin 𝜙 > 0, cos 𝜙 < 0
2
2
𝑥
−
𝑦−5 2=0
What quadrant is
2
2
this angle in?
𝑝
tan 𝜙 = −
𝐴, 𝐵, 𝐶 are not fractions
𝐴>0
10
2
=
10 2
=5 2
2
𝑥 cos 𝜙 + 𝑦 sin 𝜙 − 𝑝 = 0
2
For the HW this
2
is acceptable,
𝜙 = 𝑡𝑎𝑛−1 −1
UNLESS it’s a
unit circle value
𝜙 = 135°, 315°
Which of these will give use the quadrant we want?
𝜙 = 315°
𝑝 = 5 2,
𝜙 = 315°
By the end of the section students will be able to write the standard form of a linear equation given the length of the normal
and the angle it makes with the x-axis and write linear equations in standard form as evidenced by a mix-match activity.
SUMMARY
1.
2.
Write the equation 3𝑥 − 2𝑦 + 7 = 0 in normal
form.
Write the standard form of the equation of a
line for which the length of the normal is 3 and
makes and angle of 135° with the positive 𝑥
− 𝑎𝑥𝑖𝑠.
By the end of the section students will be able to write the standard form of a linear equation given the length of the normal
and the angle it makes with the x-axis and write linear equations in standard form as evidenced by a mix-match activity.
SUMMARY
1.
Write the equation 3 𝑥 − 2 𝑦 + 7 = 0 in normal form.
𝐴
𝐵
𝐶
± 32 + −22 = ± 13 → −
3
2
13
𝑜𝑝𝑝
𝑠𝑖𝑔𝑛
𝑜𝑓 𝐶
7
𝑥−
𝑦+
=0
− 13
− 13
− 13
3 13
2 13
7 13
−
𝑥+
𝑦−
=0
13
13
13
2.
Write the standard form of the equation of a line for
which the length of the normal is 3 and makes and
angle of 135° with the positive 𝑥 − 𝑎𝑥𝑖𝑠.
𝑥 cos 135° + 𝑦 sin 135° − 3 = 0
2
2
−
𝑥+
𝑦−3=0
2
2
− 2𝑥 + 2𝑦 − 6 = 0