Slope, Line and Angle Between Two Lines |Analytic Geometry|

Summary

This 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

00:00

📚 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.

05:03

🔍 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.

10:04

📐 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.

Keywords

💡Slope

Slope is a fundamental concept in the video, defined as the ratio of the change in the y-coordinate to the change in the x-coordinate between two points on a line, which can be mathematically expressed as \( m = \frac{y_2 - y_1}{x_2 - x_1} \). It is crucial for determining the angle of inclination of a line, which is central to the video's theme of finding the angle between two lines.

💡Angle of Inclination

The angle of inclination, or theta, is the angle a line makes with the positive direction of the x-axis. In the video, it is related to the slope through the tangent function, where \( \tan(\theta) = m \). This concept is essential for understanding how to calculate the angle between two lines.

💡Tangent

The tangent function, often abbreviated as 'tan', is used in the video to relate the slope of a line to its angle of inclination. It is a trigonometric function that calculates the ratio of the opposite side to the adjacent side in a right-angled triangle, which is analogous to the slope of a line.

💡Arctan (Arctangent)

Arctan, or arctangent, is the inverse function of the tangent and is used in the video to find the angle of inclination from the slope. It is represented as \( \theta = \arctan(m) \) and is key in deriving the angle between two lines without directly using the formula.

💡Parallel Lines

In the context of the video, parallel lines are two lines that will never intersect and have the same slope. The concept is used to illustrate a condition where lines do not form an angle with each other, contrasting with the main theme of finding the angle between intersecting lines.

💡Perpendicular Lines

Perpendicular lines are highlighted in the video as lines that intersect at a right angle, with slopes that are negative reciprocals of each other. This concept is important for understanding the relationship between the slopes of two lines that form a 90-degree angle.

💡Trigonometry

Trigonometry is the branch of mathematics that deals with the relationships between the sides and angles of triangles, particularly right-angled triangles. In the video, trigonometric principles are used to derive the formula for the angle between two lines.

💡Point-Slope Form

The point-slope form is a method of defining a line given a point on the line and its slope. In the video, it is mentioned as an alternative to the slope-intercept form, and it is used to express the relationship between a point on a line and its slope.

💡Slope-Intercept Form

The slope-intercept form is a way to represent a line in the coordinate plane, given by \( y = mx + b \), where 'm' is the slope and 'b' is the y-intercept. The video explains how to derive the slope from this form, which is necessary for calculating the angle between two lines.

💡Two-Point Form

The two-point form is another way to represent a line when two points on the line are known. In the video, it is used to calculate the slope of a line without explicitly stating it, by using the coordinates of the two points.

💡Intercept Form

The intercept form is a representation of a line where the x-intercept and y-intercept are known. In the video, it is mentioned in the context of finding the equation of a line when its intercepts are given, which can be useful for further analysis of the line's properties.

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