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# slope

The slope of a straight line in the $xy$-plane expresses how great is the change of the ordinate $y$ of the point of the line per a unit-change of the abscissa $x$ of the point; it requires that the line is not vertical.

The slope $m$ of the line may be determined by taking the changes of the coordinates between two arbitrary points $(x_{1},\,y_{1})$ and $(x_{2},\,y_{2})$ of the line:

$m=\frac{y_{2}-y_{1}}{x_{2}-x_{1}}$ |

The equation of the line is

$y=mx+b,$ |

where $b$ indicates the intersection point of the line and the $y$-axis (one speaks of y-intercept).

The slope is equal to the tangent of the slope angle of the line.

Two non-vertical lines of the plane are parallel if and only if their slopes are equal.

Keywords:

parallelism of lines

Related:

Derivative, ExampleOfRotationMatrix, ParallellismInEuclideanPlane, SlopeAngle, LineInThePlane, DifferenceQuotient, DerivationOfWaveEquation, IsogonalTrajectory, TangentOfHyperbola

Synonym:

angle coefficient (?)

Type of Math Object:

Definition

Major Section:

Reference

Parent:

Groups audience:

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## Comments

## What is "suuntakulma" in English?

Hi, has the angle which a line in the xy-plane forms with the positive x-axis some own name? I have used in a pair of entries the name "direction angle", which is a direct translation from Finnish. (The Germans speak of "Steigungswinkel" or "Neigungswinkel.)

Jussi

## Re: What is "suuntakulma" in English?

Jussi,

Since no one else has gotten back to you on this, I will put in my two cents.

To the best of my knowledge, there is no English word or phrase that is commonly used as an equivalent of the Finnish "suuntakulma". I know that, given an angle, one speaks of the ray eminating from the origin at that angle as a "terminal ray", but I have never heard of anyone wanting to do this the other way around. On the other hand, by way of analogy, I would like to suggest "terminal angle". (Even though the given ray/line might not pass through the origin, a horizontal shift can be applied, and this would not change the angle.)

I think I would have known what you meant by "direction angle" without telling me, so maybe it would be okay to use that.

Warren

## Re: What is "suuntakulma" in English?

According to this dictionary:

http://efe.scape.net/

(3) Results for finnish word: "suuntakulma"

1 suuntakulma bearing angle

2 suuntakulma direction angle

3 suuntakulma orientation angle

## Re: What is "suuntakulma" in English?

The term "bearing angle" sounds too specialized for me. The term "bearing" is used in aviation and other forms of navigation, which is probably where the phrase "bearing angle" comes from. This may also cause some confusion since, in aviation and navigation, angles are measured clockwise from the positive y axis.

Google returns nothing of mathematical significance for "orientation angle". On the other hand, for "direction angle", I found something right away. The Merriam-Webster online dictionary has:

> direction angle

> Function: noun

> an angle made by a given line with an axis of reference; specifically: such an angle made by a straight line with the three axes of a rectangular Cartesian coordinate system -- usually used in plural

If you want to see for yourself, here is the link:

http://www.m-w.com/dictionary/direction+angle

According to this definition, if you declare the "axis of reference" is the positive x axis, then "direction angle" is perfectly fine.

Apparently, "terminal angle" is in usage, but not too much. Moreover, the sites that Google returned with some (albeit minor) mathematical significance only discuss terminal angles and provide no definition or pictures.

So I guess "direction angle" wins!

Warren

## Re: What is "suuntakulma" in English?

I agree. But why not ``slope angle''? Jussi wrote:

"Hi, has the angle which a line in the xy-plane forms with the positive x-axis some own name?"

## Re: What is "suuntakulma" in English? Answer: slope angle

McGraw Hill, which is a well-known publisher of textbooks, supplies this definition (at http://www.answers.com/topic/slope-angle ):

slope angle

(mathematics) The angle of inclination of a line in the plane, where this angle is measured from the positive x axis to the line in the counterclockwise direction.

Not only is it an exact match (no need to specify x axis), but the name of the term relates well to "slope".

I am so intrigued by this definition that I think that I should add it (in my own words of course) to PM. I am assuming from this page:

http://www.answers.com/main/cite_this_answer.jsp?resource=McGraw-Hill%20...

...that their terms of usage allow me to do this, so long as I cite the indicated source. If this is not legit, please let me know.

Warren

## Re: What is "suuntakulma" in English? Answer: slope angle

Thank you Pedro and Warren, you have gone a lot of trouble to find an English term for that angle. (I found still some other alternatives: http://www.dict.cc/deutsch-englisch/Neigungswinkel.html). I agree that "slope angle" looks the best.

It however is problematic, since it is measured anticlockwise from the positive x-axis: then the descending lines get an obtuse slope angle (and it seems that such a line goes from right to left, not descends!); it means that the obtuse slope angle cannot be obtained from the usual form

arctan(m)

where m is the slope of the line. I have the view that the angle is usually defined to satisfy -90^o < angle =< 90^o. E.g. in the Finnish textbooks one says that the "suuntakulma" is the acute angle which the line forms with the x-axis, equipped with minus sign if the line descends to the right (this covers not the cases 0^o and 90^o).

Jussi

## Re: What is "suuntakulma" in English? Answer: slope angle

> (I found still some other alternatives: http://www.dict.cc/deutsch-englisch/Neigungswinkel.html).

Most of the terms listed there are not what you are looking for. There are only two of them which seem viable.

The terms "angle of inclination" and "angle of declination" make a lot of sense. In fact, the definition of "slope angle" in the source that I cited uses the phrase "angle of inclination", which I chose to drop. An angle of inclination is measured counterclockwise from the positive x axis, and an angle of declination is measured clockwise from the positive x axis.

Thus, if you wanted to use the convention that -90 deg<angle<90 deg as for "suuntakulma", you would use "angle of inclination" for lines with positive slope and "angle of declination" for lines with negative slope. As you pointed out, this convention runs into a problem though (similar problem as "slope angle"): What about lines that are parallel to the x axis?

Apparently, "angle of inclination" and "angle of declination" are not defined on PM. If you want to use this convention, these terms should be defined first! In any case, if no one else gets around to it, then I will add these. It may be a while, as I have PM projects that are more pressing (at least to me), such as the completion of the "symmetry" entry and reconstructing the "Euclidean field" (pun intended). All of these things will have to wait a while, as I will not have computer access for a few days.

Warren

## Re: What is "suuntakulma" in English? Answer: slope angle

> as you pointed out, this convention runs into a problem though (similar problem as "slope angle"): What about lines that are parallel to the x axis?

I was mistaken, choosing the angle via the below contract, one gets all values up to 90^o (cf. the entry "angle between two lines"):

> textbooks one says that the "suuntakulma" is the acute angle which the line forms with the x-axis, equipped with minus sign if the line descends to the right

I tilt to the conviction that such a definition is the best. What is the most important point, is that we could use the formula

\alpha = \arctan{m}.

Jussi

## Re: What is "suuntakulma" in English? Answer: slope angle

Correction of my preceding message:

===================================

> I was mistaken, choosing the angle via the below contract, one gets all values up to 90^o (cf. the entry "angle between two lines"):

> textbooks one says that the "suuntakulma" is the _acute_ angle which the line forms with the x-axis, equipped with minus sign if the line descends to the right

"acute" is wrong, it must be "least"! Then one gets all values up to 90^o.

Jussi