union of two straight lines through origin

[ice.rock]

hi
i would like to know that when we say that homogeneous equation represents two lines through origin.
and when are proving that we take two equations of line and say the UNION of two lines is , then multiply the equations of two lines.
I understand the concept of union of sets etc but idont understand here that when we have to take union of two lines, why multiply their equations, i totally fail to understand the logic behind that. I want to know bit of a background information for my better understanding of concept.
all help is much appreciated

 

[stapel]

i would like to know that when we say that homogeneous equation represents two lines through origin.

You would like to know that, when we say (whatever),... what? Your sentence is not complete.

and when are proving that we take two equations of line and say the UNION of two lines is , then multiply the equations of two lines.

And when (who all?) are proving that we (do stuff and say stuff), then... um... what? Your sentence does not seem to make sense.

I understand the concept of union of sets etc but idont understand here that when we have to take union of two lines, why multiply their equations, i totally fail to understand the logic behind that.

What do you mean by "taking the union of two lines"? What do you mean by "multiplying their equations"? :shock:

 

[ice.rock]

stapel with such analyzing skills you still failed to see the main points, "homogeneous equation, two straight lines". The irony.

Those interested in giving serious replies, here's an example, consider example 3 on this page.
http://www.amsi.org.au/ESA_Senior_Years/SeniorTopic1/1b/1b_3links_2.html
sorry troubling you with links. cant post the images.

Here's another link with the derivation. Please look at first two equations, when we multiply the two lines equations, the point i want clarification to. The source iwas consulting said we take union of straight lines and then multiplied the equations. If there's a logic that works behind that and is known to someone, then please share.

 

[HallsofIvy]

I don't think you understand what Stapel was saying.

However, any straight line, in an xy-coordinate system, can be written in the form ax+ by= c. If the line passes through (0, 0) then when x= 0, y=0. So a(0)+ b(0)= 0= c. That's why any line through the origin has equation ax+by= 0 (is "homogeneous"). Any point, (x,y), lying on that line, must satisfy ax+ by= 0. Any point, (x, y), on a different line through the origin satisfies a'x+ b'y= 0 for some a' and b'.

Now for any numbers, p and q, p(0)= 0 and (0)q= 0 so if a point (x, y) lies on either line (ax+ by)(a'x+ b'y)=0. Further if ab= 0, either a= 0 or b= 0 so if (x, y) satisfies (ax+ by)(a'x+ b'y)= 0 then it must satisfy either ax+ by= 0 or a'x+ b'y= 0. That is, it must lie on one or the other of the two lines.

 

[soroban]

Hello, ice.rock!

Maybe this will clear up the mystery . . . maybe not.


If I asked you to graph .$\displaystyle 6x^2 - 5xy + y^2 \:=\:0$
. . you might wonder if it is a circle, ellipse, parabola, etc.

You would find that it factors: .$\displaystyle (2x - y)(3x - y) \:=\:0$

And we have: .$\displaystyle \begin{Bmatrix}2x-y \:=\:0 \\ 3x-y \:=\:0\end{Bmatrix} \quad\Rightarrow\quad \begin{Bmatrix}y \:=\:2x \\ y \:=\:3x \end{Bmatrix}$ . . . a pair of lines.


Does that help?

 

[Bob Brown MSEE]

See Soroban's example

hi
i would like to know that when we say that homogeneous equation represents two lines through origin.
and when are proving that we take two equations of line and say the UNION of two lines is , then multiply the equations of two lines.
I understand the concept of union of sets etc but idont understand here that when we have to take union of two lines, why multiply their equations, i totally fail to understand the logic behind that. I want to know bit of a background information for my better understanding of concept.
all help is much appreciated

Soroban is saying that there is a set of points S that solve the Quadratic.

There is a set of points L1 that solve line 2x-y=0
There is a set of points L2 that solve line 3x-y=0

The product (2x-y)(3x-y)=0 is the Quadratic.
Therefore S is the Union of sets L1 and L2

 

[ice.rock]

Thanx hallsofivy, sorroban and all for the help offered. @dennis no idont even know that guy. Just made the nick like that.
I should have been more clear in asking the question , i mistankenly missed posting the link i mentioned in my last post about the proof that homogeneous equation of deg 2 represents two straight lines through origin so iam posting here, though i think at this point it's not relevant. http://spssmaths.tripod.com/cg_line_2.htm

I understood the factorization part and everything , but i was stuck with use of term union of lines . We know through elementary geo every figure including line is a set of points. So here in the beginning of proof it was written we take union of two lines and then multiplied the equations that obv resulted in a homogeneous equation. What i wanted to know this term union means taking union of all points on line 1 and line 2 and making a set or it was used to describe the product or the joining of two equations. Additionally, if it really is the union of all points of two sets i failed to understand how the product of equations gives us union.

I took this sentence from wikipedia : "In modern geometry certain degenerate cases, such as the union of two lines, are included as conics as well"

Soroban is saying that there is a set of points S that solve the Quadratic.

There is a set of points L1 that solve line 2x-y=0
There is a set of points L2 that solve line 3x-y=0

The product (2x-y)(3x-y)=0 is the Quadratic.
Therefore S is the Union of sets L1 and L2

Bob you seem really close by, but can you please explain it a little further , can you please explain that in this context:

[FONT=Times New Roman,Times]y = m[SIZE=-1]1[/SIZE]x, y = m[SIZE=-1]2[/SIZE]x[/FONT]
	[FONT=Times New Roman,Times](y -m[SIZE=-1]1[/SIZE]x)(y -m[SIZE=-1]2[/SIZE]x) = 0[/FONT]
[FONT=Arial,Helvetica]i.e.[/FONT]	[FONT=Times New Roman,Times]m[SIZE=-1]1[/SIZE]m[SIZE=-1]2[/SIZE]x[SIZE=-1]2[/SIZE]- (m[SIZE=-1]1[/SIZE] + m[SIZE=-1]2[/SIZE])xy + y[SIZE=-1]2[/SIZE] = 0 [/FONT][FONT=Arial,Helvetica]---(**)[/FONT]

[FONT=Arial,Helvetica]-- a homogenous equation of the second degree in [/FONT][FONT=Times New Roman,Times]x[/FONT][FONT=Arial,Helvetica] and [/FONT][FONT=Times New Roman,Times]y[/FONT]​
[FONT=Arial,Helvetica]The standard form of a pair of straight lines is[/FONT]
	[FONT=Times New Roman,Times]ax[SIZE=-1]2[/SIZE] + 2hxy + by[SIZE=-1]2[/SIZE] = 0 [/FONT][FONT=Arial,Helvetica]---[/FONT]

Thankyou