Perpendicular slope of 0

Perpendicular slope of 0 DEFAULT

Straight lines - product of slope of perpendicular lines.

As per wikipedia

"Two lines are parallel if and only if their slopes are equal and they are not the same line (coincident) or if they both are vertical and therefore both have undefined slopes. Two lines are perpendicular if the product of their slopes is −1 or one has a slope of 0 (a horizontal line) and the other has an undefined slope (a vertical line)."

So, it is obvious that the rule of multiplication of slopes is not applied when one of the line is vertical and another horizontal. Further, y-axis (or more specifically, the line x = 0) doesn't have a slope of infinity but it is undefined.

answered Dec 19 '16 at

Kanwaljit SinghKanwaljit Singh

8, gold badge77 silver badges bronze badges

$\endgroup$Sours: https://math.stackexchange.com/questions//straight-lines-product-of-slope-of-perpendicular-lines

Parallel & Perpendicular Lines

Purplemath

Parallel lines and their slopes are easy. Since slope is a measure of the angle of a line from the horizontal, and since parallel lines must have the same angle, then parallel lines have the same slope — and lines with the same slope are parallel.

Perpendicular lines are a bit more complicated.

If you visualize a line with positive slope (so it's an increasing line), then the perpendicular line must have negative slope (because it will have to be a decreasing line). So perpendicular lines have slopes which have opposite signs.

MathHelp.com

Need a personal math teacher?

The other "opposite" thing with perpendicular slopes is that their values are reciprocals; that is, you take the one slope value, and flip it upside down. (This is the non-obvious thing about the slopes of perpendicular lines.) Put this together with the sign change, and you get that the slope of a perpendicular line is the "negative reciprocal" of the slope of the original line — and two lines with slopes that are negative reciprocals of each other are perpendicular to each other.

To give a numerical example of "negative reciprocals", if the one line's slope is , then the perpendicular line's slope will be . Or, if the one line's slope is m = –2, then the perpendicular line's slope will be . (Remember that any integer can be turned into a fraction by putting it over 1.)


In your homework, you will probably be given some pairs of points, and be asked to state whether the lines through the pairs of points are "parallel, perpendicular, or neither". To answer the question, you'll have to calculate the slopes and compare them. Here's how that works:

  • One line passes through the points (–1, –2) and (1, 2); another line passes through the points (–2, 0) and (0, 4). Are these lines parallel, perpendicular, or neither?

To answer this question, I'll find the two slopes. So I can keep things straight and tell the difference between the two slopes, I'll use subscripts.

Since these two lines have identical slopes, then:

these lines are parallel.


  • One line passes through the points (0, –4) and (–1, –7); another line passes through the points (3, 0) and (–3, 2). Are these lines parallel, perpendicular, or neither?

I'll find the values of the slopes.

If I were to convert the "3" to fractional form by putting it over "1", then flip it and change its sign, I would get " . This negative reciprocal of the first slope matches the value of the second slope. In other words, these slopes are negative reciprocals, so:

the lines are perpendicular.


  • One line passes through the points (–4, 2) and (0, 3); another line passes through the points (–3, –2) and (3, 2). Are these lines parallel, perpendicular, or neither?

I'll find the slopes.

These slope values are not the same, so the lines are not parallel. The slope values are also not negative reciprocals, so the lines are not perpendicular. Then the answer is:


  • Find the slope of a line perpendicular to the line y = –4x + 9.

They've given me the original line's equation, and it's in "y=" form, so it's easy to find the slope. I can just read the value off the equation: m = –4.

This slope can be turned into a fraction by putting it over 1, so this slope can be restated as:

To get the negative reciprocal, I need to flip this fraction, and change the sign. Then the slope of any line perpendicular to the given line is:

Warning: When asked a question of this type ("are these lines parallel or perpendicular?"), do not start drawing pictures. If the lines are close to being parallel or close to being perpendicular (or if you draw the lines messily), you can very-easily get the wrong answer from your picture.

Besides, they're not asking if the lines look parallel or perpendicular; they're asking if the lines actually are parallel or perpendicular. The only way to be sure of your answer is to do the algebra.


URL: https://www.purplemath.com/modules/slope3.htm

Sours: https://www.purplemath.com/modules/slope3.htm
  1. D3 micro chest rig
  2. Ultrasonic deer repellent reviews
  3. Wolf drawing reference

Intermediate Algebra
Tutorial The Slope of a Line


WTAMU> Virtual Math Lab > Intermediate Algebra  



deskLearning Objectives


 
After completing this tutorial, you should be able to:
  1. Find the slope given a graph, two points or an equation.
  2. Write a linear equation in slope/intercept form.
  3. Determine if two lines are parallel, perpendicular, or neither. 




desk Introduction



This tutorial takes us a little deeper into linear equations.  We will be looking at the slope of a line.  We will also look at the relationship between the slopes of parallel lines as well as perpendicular lines.  Let's see what you can do with slopes.

 

 

desk Tutorial



 


 

The slope of a line measures the steepness of the line.

Most of you are probably familiar with associating slope with "rise over run". 
 

Rise means how many units you move up or down from point to point.  On the graph that would be a change in the y values.

Run means how far left or right you move from point to point.  On the graph, that would mean a change of x values.


 

Here are some visuals to help you with this definition:
 

Positive slope:

positive slope

positive slope

Note that when a line has a positive slope it goes up left to right.


 

Negative slope:

negative slope

negative slope

Note that when a line has a negative slope it goes down left to right.


 

Zero slope:

zero slope

slope = 0 

Note that when a line is horizontal the slope is 0.


 

Undefined slope:

undefined slope

slope = undefined 

Note that when the line is vertical the slope is undefined.


 
 
 
 
Slope Formula Given Two Points

Given two points x 1 and x 2

slope formula


 

The subscripts just indicate that these are two different points.  It doesn't matter which one you call point 1 and which one you call point 2 as long as you are consistent throughout that problem. 

Note that we use the letter m to represent slope. 

notebook Example 1: Find the slope of the straight line that passes through (-5, 2) and (4, -7).


 
example 1


*Plug in x and y values into slope formula

*Simplify
 
 

 


 

Make sure that you are careful when one of your values is negative and you have to subtract it as we did in line 2.  4 - (-5) is not the same as 4 - 5. 

The slope of the line is


 
 
notebook Example 2: Find the slope of the straight line that passes through  (1, 1) and (5, 1).

 
example 2

*Plug in x and y values into slope formula

*Simplify
 

 


 

It is ok to have a 0 in the numerator.  Remember that 0 divided by any non-zero number is 0.
 

The slope of the line is 0.


 
 
notebook Example 3: Find the slope of the straight line that passes through (3, 4) and (3, 6).

 
example 3

*Plug in x and y values into slope formula

*Simplify

 


 

Since we did not have a change in the x values, the denominator of our slope became 0.  This means that we have an undefined slope.  If you were to graph the line, it would be a vertical line, as shown above.

The slope of the line is undefined.


 

Slope/Intercept Equation of a Line

slope intercept form


 

If your linear equation is written in this form, m represents the slope and b represents the y-intercept.

This form can be handy if you need to find the slope of a line given the equation.


 
 
 
Function Notation of the 
Slope/Intercept Equation of a Line

slope intercept


 
m still represents slope and b still represents the y-intercept. 

 
 
notebook Example 4:    Find the slope and the y-intercept of the line example 4a.

 

As mentioned above, if the equation is in the slope/intercept form, we can easily see what the slope and y-intercept are. 
 

Let&#;s go ahead and get it into the slope/intercept form first:


 
example 4b


*Sub. 3x and add 6 to both sides

*Inverse of mult. by 3 is div. by 3

*Written in slope/intercept form
 


 

Lining up the form with the equation we got, can you see what the slope and y-intercept are?

In this form, the slope is m, which is the number in front of x.  In our problem, that would have to be  

In this form, the y-intercept is b, which is the constant.  In our problem, that would be 2. 
 

The answer is the slope is -1 and the y-intercept is 2.


 
 
 
notebook Example 5:    Find the slope and the y-intercept of the line example 5a.

 

This example is written in function notation, but is still linear.  As shown above, you can still read off the slope and intercept from this way of writing it. 

In this example, it is already written in the slope/intercept form, so we do not have to mess around with it.  We can get down to business and answer our question of what are the slope and y-intercept.


 
example 5b


*Written in slope/intercept form


 

Lining up the form with the equation we got, can you see what the slope and y-intercept are?

In this form, the slope is m, which is the number in front of x.  In our problem, that would have to be 2. 

In this form, the y-intercept is b, which is the constant.  In our problem, that would be  
 

The answer is the slope is 2 and the y-intercept is


 
notebook Example 6:    Find the slope and the y-intercept of the line  x= 5.

 

Note how we do not have a y.  This type of linear equation was shown in Tutorial Graphing Linear Equations.  When we have x = c, where c is a constant, then this graph is what type of line?
If you said vertical, you are correct. 

Since this is a special type of linear equation that can&#;t be written in the slope/intercept form, I&#;m going to give you a visual of what is happening and then from that let&#;s see if we can&#;t figure out the slope and y-intercept.

The graph would look like this:

example 6a


 

First, let&#;s talk about the slope.  Note that all the x values on this graph are 5.  That means the change in x, which is the denominator of the slope formula, would be 5 - 5 = 0.  Well you know that having a 0 in the denominator is a big no, no.  This means the slope is undefined.  As shown above, whenever you have a vertical line your slope is undefined.

Now let&#;s look at the y-intercept.  Looking at the graph, you can see that this graph never crosses the y-axis, therefore there is no y-intercept either.  Another way to look at this is the x value has to be 0 when looking for the y-intercept and in this problem x is always 5.

So, for all our efforts on this problem, we find that the slope is undefined and the y-intercept does not exist.


 
 
 
notebook Example 7:    Find the slope and the y-intercept of the line y=

 

Note how we do not have an x.  This type of linear equation was shown in Tutorial Graphing Linear Equations.  When we have y = c, where c is a constant, then this graph is what type of line?
If you said horizontal, you are correct. 

Since this is a special type of linear equation that can&#;t be written in the slope/intercept form, I&#;m going to give you a visual of what is happening and then from that let&#;s see if we can&#;t figure out the slope and y-intercept.
 

The graph would look like this:

example 7a


 

First, let&#;s talk about the slope.  Note how all of the y values on this graph are   That means the change in y, which is the numerator of the slope formula would be -2 - (-2) = 0. Having 0 in the numerator and a non-zero number in the denominator means only one thing.  The slope equals 0.

Now let&#;s look at the y-intercept.  Looking at the graph, you can see that this graph crosses the y-axis at (0, -2).  So the y-intercept is (0, -2).

The slope is 0 and the y-intercept is


 

Parallel Lines and Their Slopes
parallel lines

 

In other words, the slopes of parallel lines are equal.

Note that two lines are parallel if there slopes are equal and they have different y-intercepts.

parallel lines


 

Perpendicular Lines and Their Slopes
perpendicular lines

 

In other words, perpendicular slopes are negative reciprocals of each other.

perpendicular lines


 
 
notebook Example 8:    Determine if the lines are parallel, perpendicular, or neither. example 8aand example 8b.

 

In order for these lines to be parallel their slopes would have to be equal and to be perpendicular they would have to be negative reciprocals of each other. 

So let&#;s find out what the slopes are.  Since the equations are already in the slope/intercept form, we can look at them and see the relationship between the slopes.  What do you think?  The slope of the first equation is 7 and  the slope of the second equation is 7

Since the two slopes are equal and their y-intercepts are different, the two lines would have to be parallel.


 
 
 
notebook Example 9:    Determine if the lines are parallel, perpendicular, or neither. example 9aand example 9b.

 

Again, the equations are already in the slope/intercept form, so let&#;s go right to looking for the slope.  What did you find? 

I found that the slope of the first equation is 4 and the slope of the second equation is -1/4.  So what does that mean? 

Since the two slopes are negative reciprocals of each other, the two lines would be perpendicular to each other. 


 
 
 
notebook Example 10:    Determine if the lines are parallel, perpendicular, or neither. example 10aand example 10b.

 

Writing the first equation in the slope/intercept form we get:


 
example 10c


*Inverse of add 10x is sub. 10x
*Written in slope/intercept form


 

Writing the second equation in the slope/intercept form we get:


 
example 10d


*Inverse of add 4x is sub. 4x
 

*Inverse of mult. by 2 is div. by 2

*Written in slope/intercept form


 

In order for these lines to be parallel their slopes would have to be equal and to be perpendicular they would have to be negative reciprocals of each other.  So let&#;s find out what the slopes are.  Since the equations are now in the slope/intercept form, we can look at them and see the relationship between the slopes.  What do you think? 

The slope of the first equation is and the slope of the second equation is  

Since the two slopes are not equal and are not negative reciprocals of each other, then the answer would be neither.


 
 
desk Practice Problems
  

These are practice problems to help bring you to the next level.  It will allow you to check and see if you have an understanding of these types of problems. Math works just like anything else, if you want to get good at it, then you need to practice it.  Even the best athletes and musicians had help along the way and lots of practice, practice, practice, to get good at their sport or instrument.  In fact there is no such thing as too much practice.

To get the most out of these, you should work the problem out on your own and then check your answer by clicking on the link for the answer/discussion for that  problem.  At the link you will find the answer as well as any steps that went into finding that answer.

 

pencil Practice Problem 1a - 1b: Find the slope of the straight line that passes through the given points.

 

 

pencil Practice Problems 2a - 2c: Find the slope and the y-intercept of the line.

 

 

 

pencil Practice Problems 3a - 3b: Determine if the lines are parallel, perpendicular, or neither.

 

 

pencil Practice Problem 4a: Determine the slope of the line.

 


desk Need Extra Help on these Topics?



 

 


WTAMU> Virtual Math Lab > Intermediate Algebra


Last revised on July 3, by Kim Seward.
All contents copyright (C) - , WTAMU and Kim Seward. All rights reserved.

Sours: https://www.wtamu.edu/academic/anns/mps/math/mathlab/int_algebra/int_alg_tut15_slope.htm

 

Slope-Intercept Form

Linear functions are graphically represented by lines and symbolically written in slope-intercept form as,

y = mx + b,

where m is the slope of the line, and b is they-intercept. We call b the y-intercept because the graph of y = mx + b intersects the y-axis at the point (0, b). We can verify this by substituting x = 0 into the equation as,

y = m · 0 + b = b.

Notice that we substitute x = 0 to determine where a function intersects the y-axis because the x-coordinate of a point lying on the y-axis must be zero.

The Definition of Slope :

The constant m expressed in the slope-intercept form of a line, y = mx + b, is the slope of the line. Slope is defined as the ratio of the rise of the line (i.e. how much the line rises vertically) to the run of line (i.e. how much the line runs horizontally).

Definition

For any two distinct points on a line, (x1, y1) and (x2, y2), the slope is,

Equation: m = (y2 - y1)/ (x2 - x1); x1 cannot equal x2

 

 

Intuitively, we can think of the slope as measuring the steepness of a line. The slope of a line can be positive, negative, zero, or undefined. A horizontal line has slope zero since it does not rise vertically (i.e. y1y2 = 0), while a vertical line has undefined slope since it does not run horizontally (i.e. x1x2 = 0).

Zero and Undefined Slope

As stated above, horizontal lines have slope equal to zero. This does not mean that horizontal lines have no slope. Since m = 0 in the case of horizontal lines, they are symbolically represented by the equation, y = b. Functions represented by horizontal lines are often called constant functions. Vertical lines have undefined slope. Since any two points on a vertical line have the same x-coordinate, slope cannot be computed as a finite number according to the formula,

equation: m = (y2 - y1)/ (x2 - x1)

because division by zero is an undefined operation. Vertical lines are symbolically represented by the equation, x = a where a is the x-intercept. Vertical lines are not functions; they do not pass the vertical line test at the point x = a.

Positive Slopes

Lines in slope-intercept form with m > 0 have positive slope. This means for each unit increase in x, there is a corresponding m unit increase in y (i.e. the line rises by m units). Lines with positive slope rise to the right on a graph as shown in the following picture,

graph

Lines with greater slopes rise more steeply. For a one unit increment in x, a line with slope
m1 = 1 rises one unit while a line with slope m2 = 2 rises two units as depicted,

graph

Negative Slopes

Lines in slope-intercept form with m < 0 have negative slope. This means for each unit increase in x, there is a corresponding |m| unit decrease in y (i.e. the line falls by |m| units). Lines with negative slope fall to the right on a graph as shown in the following picture,

graph

The steepness of lines with negative slope can also be compared. Specifically, if two lines have negative slope, the line whose slope has greatest magnitude (known as the absolute value) falls more steeply. For a one unit increment in x, a line with slope m3 = −1 falls one unit while a line with slope m4= −2 falls two units as depicted,

graph

Parallel and Perpendicular Lines

Two lines in the xy-plane may be classified as parallel or perpendicular based on their slope. Parallel and perpendicular lines have very special geometric arrangements; most pairs of lines are neither parallel nor perpendicular. Parallel lines have the same slope. For example, the lines given by the equations,

y1 = −3x + 1,

y2 = −3x − 4,

are parallel to one another. These two lines have different y-intercepts and will therefore never intersect one another since they are changing at the same rate (both lines fall 3 units for each unit increase in x). The graphs of y1 and y2 are provided below,

graph

Perpendicular lines have slopes that are negative reciprocals of one another. In other words, if a line has slope m1, a line that is perpendicular to it will have slope,

equation: m2 + -(1/m1)

An example of two lines that are perpendicular is given by the following,

Equation 1: y3 = 2x + 3, y4 = -1/2

These two lines intersect one another and form ninety degree (90°) angles at the point of intersection. The graphs of y3 and y4 are provided below,

 

graph

*****

In the next section we will describe how to solve linear equations.

Linear equations

Sours: http://www.biology.arizona.edu/biomath/tutorials/linear/slope.html

0 of perpendicular slope

DMCA Complaint

If you believe that content available by means of the Website (as defined in our Terms of Service) infringes one or more of your copyrights, please notify us by providing a written notice (“Infringement Notice”) containing the information described below to the designated agent listed below. If Varsity Tutors takes action in response to an Infringement Notice, it will make a good faith attempt to contact the party that made such content available by means of the most recent email address, if any, provided by such party to Varsity Tutors.

Your Infringement Notice may be forwarded to the party that made the content available or to third parties such as ChillingEffects.org.

Please be advised that you will be liable for damages (including costs and attorneys’ fees) if you materially misrepresent that a product or activity is infringing your copyrights. Thus, if you are not sure content located on or linked-to by the Website infringes your copyright, you should consider first contacting an attorney.

Please follow these steps to file a notice:

You must include the following:

Send your complaint to our designated agent at:

Charles Cohn Varsity Tutors LLC
S. Hanley Rd, Suite
St. Louis, MO

Or fill out the form below:

 

Sours: https://www.varsitytutors.com/algebra_1-help/how-to-find-the-slope-of-perpendicular-lines
Perpendicular Line Slope

DMCA Complaint

If you believe that content available by means of the Website (as defined in our Terms of Service) infringes one or more of your copyrights, please notify us by providing a written notice (“Infringement Notice”) containing the information described below to the designated agent listed below. If Varsity Tutors takes action in response to an Infringement Notice, it will make a good faith attempt to contact the party that made such content available by means of the most recent email address, if any, provided by such party to Varsity Tutors.

Your Infringement Notice may be forwarded to the party that made the content available or to third parties such as ChillingEffects.org.

Please be advised that you will be liable for damages (including costs and attorneys’ fees) if you materially misrepresent that a product or activity is infringing your copyrights. Thus, if you are not sure content located on or linked-to by the Website infringes your copyright, you should consider first contacting an attorney.

Please follow these steps to file a notice:

You must include the following:

Send your complaint to our designated agent at:

Charles Cohn Varsity Tutors LLC
S. Hanley Rd, Suite
St. Louis, MO

Or fill out the form below:

 

Sours: https://www.varsitytutors.com/act_math-help/how-to-find-the-slope-of-a-perpendicular-line

You will also like:

How do you find the slope of a line perpendicular to another line?

How do you find the slope of a line perpendicular to another line?

To find the slope of the given line we need to get the line into slope-intercept form (y = mx + b), which means we need to solve for y: The slope of the line 4x – 5y = –10 is m = 4/5. Therefore, the slope of the line perpendicular to this line would have to be m = –5/4.

What is the slope of a line perpendicular to this line what is the slope of a line parallel to this line?

Vertical lines and horizontal lines are perpendicular to each other. The slope of the perpendicular line in this case would be the slope of a horizontal line which would be 0. The slope of the parallel line is undefined and the slope of the perpendicular line is 0.

What slope is perpendicular to 8?

Using the slope-intercept form, the slope is Undefined. The slope of a perpendicular line to a vertical line is zero.

What is the slope of a line that is perpendicular to the line shown on the graph 4?

If the line has a slope of 4, then a line perpendicular to it would have a slope of -1/4. If the line has a slope of -4, then a line perpendicular to it would have a slope of 1/4.

Which line is perpendicular to a line that has a slope of 1 3?

The slope of a line perpendicular to one with the slope of 13 is −3 . See explanation.

What is the slope of a line that is perpendicular to the line y x 4 the slope of the line is?

Answer Expert Verified The opposite reciprocal of 1 is -1, thus the slope of a line perpendicular to the line y = x + 4 is Hope this helps.

What is the slope of a line perpendicular to Y =- 7 4x?

Using the slope-intercept form, the slope is −74 . The equation of a perpendicular line to y=−7&#;4 y = &#; 7 x 4 must have a slope that is the negative reciprocal of the original slope. Simplify the result. Cancel the common factor of 1 1 and −1 &#; 1 .

What is the slope of a line that is perpendicular to the line y 1 6x 4 the slope of the line is?

The slope would be -6 because it is the opposite reciprocal.

What is the slope of a line that is perpendicular to the line y 2x 1?

Algebra Examples Using the slope-intercept form, the slope is 2 2 .

What is the perpendicular slope of 0?

The slope of a line perpendicular to a line with a slope of 0 is undefined. The perpendicular line is a vertical line. A line with a slope of 0 is a horizontal line.

What happens if the slope of a line is 0?

Since we did not have a change in the x values, the denominator of our slope became 0. This means that we have an undefined slope. If you were to graph the line, it would be a vertical line, as shown above. The slope of the line is undefined.

What does parallel lines look like?

Parallel lines look like railroad tracks: they are always the same distance apart, running next to each other. The lines do intersect. Next, determine if the lines intersect at a right angle. The lines do not intersect at a right angle.

What is the slope of the line 0?

The slope of a line can be positive, negative, zero, or undefined. A horizontal line has slope zero since it does not rise vertically (i.e. y1 − y2 = 0), while a vertical line has undefined slope since it does not run horizontally (i.e. x1 − x2 = 0). because division by zero is an undefined operation.

Is a line with a slope of 0 linear?

No, so its slope can&#;t be positive. This relationship always holds: a slope of zero means that the line is horizontal, and a horizontal line means you&#;ll get a slope of zero. (By the way, all horizontal lines are of the form &#;y = some number&#;, and the equation &#;y = some number&#; always graphs as a horizontal line.)

What does a zero slope mean?

A slope of zero means that there is a constant relationship between x and y. Graphically, the line is flat; the rise over run is zero.

What is the slope intercept form if the slope is 0?

y=mx + b When the equation is in this form, the coefficient of x will be the slope and the y-intercept will be (0,b). If the denominator of the fraction is 0, the slope is undefined. This occurs if the x value is the same for both points.

22/05/Manon WilcoxPopular articles

Sours: https://colors-newyork.com/how-do-you-find-the-slope-of-a-line-perpendicular-to-another-line/


5682 5683 5684 5685 5686