Slope, Line and Angle Between Two Lines |Analytic Geometry|
Summary
TLDRThis video tutorial offers a quick method for calculating the angle between two lines using the concept of slope. It explains the relationship between slope and angle of inclination, and how to determine if lines are parallel or perpendicular based on their slopes. The main formula for finding the angle between two intersecting lines is provided, along with an alternative strategy that involves understanding the tangent of the angle as the difference in slopes. The video also covers different forms of line equations, including point-slope and slope-intercept forms, to aid in the calculation process. The presenter concludes with an example to illustrate the application of these concepts.
Takeaways
- 📚 The video teaches how to find the angle between two lines using the fastest method.
- 📐 The slope (m) of a line is calculated by the formula \( m = \frac{y_2 - y_1}{x_2 - x_1} \).
- 📈 The angle of inclination (θ) of a line is related to its slope through the tangent function, \( \tan(\theta) = m \).
- 🔄 If two lines are parallel, they have the same slope.
- ⊥ If two lines are perpendicular, one line's slope is the negative reciprocal of the other's.
- 🤔 The formula to find the angle between two intersecting lines with slopes m1 and m2 is \( \theta = \arctan\left(\frac{m2 - m1}{1 + m1 \cdot m2}\right) \).
- 💡 An alternate solution to finding the angle between two lines is by considering the difference between the angles of inclination of each line.
- 📝 The point-slope form of a line is \( y - y_1 = m(x - x_1) \).
- 📑 The slope-intercept form of a line is \( y = mx + b \), where m is the slope and b is the y-intercept.
- 📍 The two-point form of a line is derived from two points and is \( y - y_1 = \frac{y_2 - y_1}{x_2 - x_1}(x - x_1) \).
- 📉 The intercept form of a line is \( \frac{x}{a} + \frac{y}{b} = 1 \), where a and b are the x and y intercepts, respectively.
- 🔢 The video provides an example of finding the angle between two lines with given equations, emphasizing the use of the slope and the arctan function.
Q & A
What is the basic concept of slope in the context of this video?
-The slope, denoted as 'm', is the ratio of the change in the y-coordinate (rise) to the change in the x-coordinate (run), mathematically represented as \( m = \frac{y_2 - y_1}{x_2 - x_1} \).
How is the angle of inclination of a line related to its slope?
-The angle of inclination, denoted as 'theta', is related to the slope through the tangent function, where \( \tan(\theta) = m \), meaning the angle is the arctangent of the slope.
What does it mean if two lines have the same slope?
-If two lines have the same slope, it means they are parallel to each other.
How can you determine if two lines are perpendicular based on their slopes?
-Two lines are perpendicular if the product of their slopes \( m_1 \times m_2 \) equals -1, i.e., \( m_1 = -\frac{1}{m_2} \).
What is the formula for finding the angle between two intersecting lines given their slopes?
-The formula to find the angle between two lines with slopes \( m_1 \) and \( m_2 \) is \( \theta = \arctan\left(\frac{m_2 - m_1}{1 + m_1m_2}\right) \).
What is an alternative method to find the angle between two lines without using the standard formula?
-An alternative method involves finding the individual angles of inclination for each line (theta1 and theta2) and then calculating the difference, \( \theta = \theta_2 - \theta_1 \).
What is the point-slope form of a line equation and how is it used in this context?
-The point-slope form is given by \( y - y_1 = m(x - x_1) \), where \( m \) is the slope and \( (x_1, y_1) \) is a point on the line. It's used to describe the line in terms of a point and its slope.
What is the slope-intercept form of a line equation and what does it represent?
-The slope-intercept form is \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. It represents the line's equation with respect to its slope and the point where it crosses the y-axis.
Can you explain the two-point form of a line equation and how it differs from the other forms?
-The two-point form is derived from two points \( (x_1, y_1) \) and \( (x_2, y_2) \) on the line and is given by \( \frac{y - y_1}{y_2 - y_1} = \frac{x - x_1}{x_2 - x_1} \). It differs from other forms as it does not explicitly require the slope to be known.
What is the intercept form of a line equation and how is it used?
-The intercept form is \( \frac{x}{a} + \frac{y}{b} = 1 \), where \( a \) and \( b \) are the x and y intercepts, respectively. It's used when the intercepts are known to describe the line without needing the slope.
How does the video suggest finding the slope of a line given its equation in standard form?
-The video suggests rearranging the standard form equation to the slope-intercept form, \( y = mx + b \), to easily identify the slope \( m \) as the coefficient of \( x \).
What is the final step in the video's method for finding the angle between two lines?
-The final step is to use the arctangent function with the calculated values to find the angle in degrees, ensuring the calculator is in degree mode.
Outlines
📚 Understanding Slope and Angle Calculation
This paragraph introduces the concept of slope and angle between two lines. It explains that the slope is the rise over run or the change in y over the change in x, represented as m = (y2 - y1) / (x2 - x1). The angle of inclination, θ, is related to the slope through the tangent function, where tan(θ) equals the slope. The paragraph also touches on the conditions for lines being parallel or perpendicular based on their slopes. Finally, it presents a formula for finding the angle between two intersecting lines using their slopes, m1 and m2, which is θ = arctan((m2 - m1) / (1 + m1*m2)). An alternative strategy without using the formula is hinted at involving the point slope form of a line.
🔍 Exploring Line Equations and Slope Calculation
The second paragraph delves into different forms of line equations, including the point slope form, slope intercept form, and two point form. It explains the point slope form as y - y1 = m(x - x1) and the slope intercept form as y = mx + b, where m is the slope and b is the y-intercept. The two point form is also described, which does not explicitly provide the slope but can be used to calculate it using two points (x1, y1) and (x2, y2). The paragraph then illustrates how to find the slope from the given line equations, using examples to demonstrate the process for both a linear equation in slope intercept form and one that needs to be rearranged into this form.
📐 Calculating the Angle Between Two Lines
In the final paragraph, the focus shifts to calculating the angle between two lines using their slopes. It provides a step-by-step approach to determine which slope is larger and then uses this information to apply the arctan function to find the angle in degrees. The example given involves two lines with slopes of -2 and -1/3, leading to the conclusion that the angle between them is 45 degrees. The paragraph emphasizes the practicality of this method and hints at further topics that will be covered in subsequent videos, including analytic geometry, calculus, and engineering mechanics.
Mindmap
Keywords
💡Slope
💡Angle of Inclination
💡Tangent
💡Arctan (Arctangent)
💡Parallel Lines
💡Perpendicular Lines
💡Trigonometry
💡Point-Slope Form
💡Slope-Intercept Form
💡Two-Point Form
💡Intercept Form
Highlights
Introduction to the method for solving the angle between two lines using the slope concept.
Explanation of slope calculation using the formula m = (y2 - y1) / (x2 - x1).
Relating the slope to the angle of inclination using the tangent function.
Condition for parallel lines: having the same slope.
Condition for perpendicular lines: slopes being negative reciprocals of each other.
Derivation of the formula for the angle between two intersecting lines using trigonometry.
Presentation of the formula for finding the angle between two lines: θ = arctan((m2 - m1) / (1 + m1 * m2)).
Alternative solution without the formula by analyzing the angles formed by the lines.
Introduction to the point-slope form of a line equation.
Explanation of the slope-intercept form and its components.
Description of the two-point form and how to derive the slope from two points.
Intercept form of a line equation and its relation to x and y intercepts.
Application of the slope-intercept form to find the slope of a given line equation.
Comparison of slopes to determine the larger one for the angle calculation.
Use of the arctan function to calculate the angle between two lines without the standard formula.
Emphasis on the importance of understanding the slope concept in analytic geometry.
Preview of upcoming topics in the series, including conic sections and calculus.
Transcripts
hi guys
in this video i will teach you on how to
solve
the angle between two lines the fastest
way so
let's recall jung slope mona so the
slope
is given by you have the slope
m is given by your rise all over run
or you have rise as change in y
pull over change in
x where your slope will be
you have y sub 2 minus y sub 1
over x sub 2 minus x sub 1.
now the angle of inclination of this
line
is given by theta here so your angle
theta you have tangent
of theta will be your opposite over
adjacent that is y sub 2
minus y sub 1 and over x sub 2 minus x
sub 1
or your slope therefore your angle theta
here is equal to your arc time
of your slope
now suppose we have two lines so if the
lines are parallel they have the same
slope
so if you have equation one and
line two line one and line two if they
are part of the
their slope are equal
the lines are tangent to each other or
perpendicular
so m1 is equal to the negative
reciprocal of
the other slope so it means that these
two lines
are tangent to each other so you can
derive that using your
trigonometry now let's say
we are required to find the angle
between two lines
so we have the formula in finding that
angle for example we have these two
lines
two intersecting lines you have here
line one
and you have here
[Applause]
line two so these two lines of course
they have slopes
say m1 and
m2 so we have the
angle between those two lines so in
various books we have the formula
in finding the angles you have tangent
of theta is equal to m2
minus m1 over
1 plus m2 times
m1 or theta is equal to
arctan of
m2 minus m1
all over 1 plus m2 times m1
so we have here our ready-made
formula so what if you forgot the
formula
so i have a technique or strategy in
solving this type of problem
we're not even using this formula so
first let's analyze the prob
uh the figure suppose we have two lines
here
you have
this is line two and this is
line one
so this is why this is x
so this angle here you have angle 1
and this angle here you have
say angle 2 so the angle between
those lines is this angle let's say
theta
so theta is actually equal to
theta sub 2 minus
this
sub 1 now what is theta sub 2 and what
is theta sub 1
so as you recall that the angle
you have tangent of angle theta is just
equal to your slope
okay it means that angle will be
part time of your
slope it means that this angle theta
is equal to our time
of slope two minus
part time of slope
one so this is our alternate
solution
is what you call the point slope form so
one of them points low form
so point slope form is a
so young point slope form is given by
you have y
minus y one equals
m times x minus
x sub one so this is the point slow form
now we have what we call the slope
intercept form
so the slope intercept form is equal is
given by y
is equal to mx
plus b so m 10 that is the slope
so bhagavati now we're gonna slope
intercept form because
you have this two then you have the
y-intercept so b
and y-intercept yeah because if x is
zero y
equals b so it means that b is your
y-intercept for example you have this
line
so this line have a slope and
then your y-intercept b
and the third you have the two point
form
so two point four it means that you are
given two points so
you have d two lambda manchego
points though form so y minus y one
equals m times x minus x sub one
zero slope is not given so it means that
b begins
for example you have
so even cannon 2 points a line
say x1 y1 then
x2 zero slope nothing so y minus y sub
one equals
you have y sub two minus y sub one all
over
x sub two minus x sub one so times x
minus
x sub one so this is your
two point form b
form now
the equation of the line is what you
call the intercept form
bibliogram x and y interception form is
equal to
sine form is given by x equals
x over e plus
y over b equals one in where a and b
are your x and y intercept
example you have this curve
so given canaan b you have the
k and connecting those points creates a
line so this is intercept form so
x over a plus y over b
equals 1 so you are given
the intercept of your line
now proceed on young line has an
important shot
in finding the slope given canal
line
the equation of linear will just will
just reduce
to your general equation of the line
so now in this problem you are required
to find the
angle form between lines now wagner and
gomez formula so the regenerators
alternate solution
so the first step is
so first step find the slope say for
equation one you have
so y equals
you have negative two x plus eight
so the slope is formula is y is equal to
m
x plus b so your m is negative
two so let's say the slope of this line
is
say equation one is negative
two now let's proceed to the second line
so that is x plus
three y plus four equals zero so
you have three y equals to negative
x minus four so divide both sides by
three
we have y equals negative
x over three minus we have four
thirds now the stop of this line
equation two or line two is equal to
you have negative one third now we
compare
which slope is larger so
we have the larger slope negative
one third so i'm waiting for nothing is
your theta
equals
so automatically pythologan formula
angle between lines so you have
arctan so be sure guys in a category
degrees so mode shift mode
you have degrees then
are tan you have negative one third
minus
so arc tan negative two
so you have 45 degrees
you don't procedure guys like in a game
with nothing it's very useful because
then you will arrive at the same answer
so that's it guys so i hope that you
learned from this
topic so next topic procedures
until machine design so continuous
analytic geometry angulong conic
sections then position
differential and integral calculus
we have the physics and thermodynamics
engineering mechanics statics and
dynamics see you in my next video
guys
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