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【数学】 直線の方程式 Line equation

A very useful equation which represents a line.

https://remedics.air-nifty.com/photos/hawaii/graph4.gif

With this form, you can tell that this line goes through point (a, b)

and parallel to

vactor (A, B).

If you are given a line formula of

https://remedics.air-nifty.com/photos/hawaii/graph3.gif,

you can automatically tell that this line goes through

point (-2, -1)

https://remedics.air-nifty.com/photos/hawaii/geogebra27.png

and parallel to

vector (3, 4)

https://remedics.air-nifty.com/photos/hawaii/geogebra28.png

https://remedics.air-nifty.com/photos/hawaii/geogebra26.png

Therefore the line is
https://remedics.air-nifty.com/photos/hawaii/geogebra24.png

Usualiy a line is expressed in a form of

https://remedics.air-nifty.com/photos/hawaii/graph5.gif.

In this form, the slope is - a/b

https://remedics.air-nifty.com/photos/hawaii/graph16.gif

Example:

For a line equation of

https://remedics.air-nifty.com/photos/hawaii/graph17.gif

This line has a slope of 4/3, and y-intercept of 5/3

https://remedics.air-nifty.com/photos/hawaii/graph18.gif.

But it is often a bit cumbersome to translate into this equation.

Sometimes it is much easyer to transform the equation,

https://remedics.air-nifty.com/photos/hawaii/graph17.gif

into

https://remedics.air-nifty.com/photos/hawaii/graph3.gif.

Then, you can say,

the line goes through a point (-2, -1) and parallel to vector (3, 4).

Particularly, this form is useful when it comes to 3D line object.



Here goes the main topic of this article.

Guess what kind of line is described by the following equation?

https://remedics.air-nifty.com/photos/hawaii/graph6_2.gif

This is a 3D object.

This is a line goes through

point (a, b, c)

and parallel to

vector (A, B, C).

https://remedics.air-nifty.com/photos/hawaii/geogebra29.png

For example

https://remedics.air-nifty.com/photos/hawaii/graph7.gif

This line goes through a

point (2, 1, 3)

https://remedics.air-nifty.com/photos/hawaii/geogebra30.png

and parallel to

vactor (3, 4, 6)

https://remedics.air-nifty.com/photos/hawaii/geogebra31.png

As such, the line is
https://remedics.air-nifty.com/photos/hawaii/geogebra32.png
Looking from the different angle.
https://remedics.air-nifty.com/photos/hawaii/geogebra33.png


Please note.

A line equation is often coming with "t" like

https://remedics.air-nifty.com/photos/hawaii/graph19.gif

or

https://remedics.air-nifty.com/photos/hawaii/graph20.gif

These are the same.

Line.

Don't be panic with the presence of "t".



If you are given formula of

https://remedics.air-nifty.com/photos/hawaii/graph22.gif

Transform it to

https://remedics.air-nifty.com/photos/hawaii/graph21.gif.

Now, you can tell they are just asking a line about

https://remedics.air-nifty.com/photos/hawaii/graph7.gif.

In this form , you can imagine the line passing through (2, 1, 3) and parallel to a vector (3, 4, 6).

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