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ganeshie8
 one year ago
how do i explain this to a 10th grader who was just introduced to lines and planes
explain why a line in 3space cannot be represented by a scalar equation like \(ax+by=C\)
ganeshie8
 one year ago
how do i explain this to a 10th grader who was just introduced to lines and planes explain why a line in 3space cannot be represented by a scalar equation like \(ax+by=C\)

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Owlcoffee
 one year ago
Best ResponseYou've already chosen the best response.4Because the "z" component is equal to zero: \[ax+by+(0)z=C\] This means that it has "z" coordinate of zero, meaning that it only defines a plane, and that plane will be created by the family of lines: \[r_1 + k(r_2)=0\]

Empty
 one year ago
Best ResponseYou've already chosen the best response.1In some sense you can, the problem is you have to use a system of linear equations to represent a line. So let's just say the z component depends on x and y, this is true for a line. \(z(x,y)=ax+by+c\) but the problem is this lets us pick any point in the xy plane to a height, which will give us an entire surface! We need to restrict what values we can pick for x and y, so we need to go further and define \(y(x)=mx+n\) which will be the projection of our actual line onto the xy plane, since we are going to map only points from a line to a line on the plane represented by \(z(x,y)\) to get our line.

ikram002p
 one year ago
Best ResponseYou've already chosen the best response.2give her a square and divide it to slices with same ax+by=c equation, so she would figure out its not unique representation.

ikram002p
 one year ago
Best ResponseYou've already chosen the best response.2if you got what i mean dw:1438078875765:dw

ikram002p
 one year ago
Best ResponseYou've already chosen the best response.2u can also use the book itself and papers as slices

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0the equation \(ax+by=C\) suggests that we're looking for points \((x,y)\) that have the same inner product with respect to \((a,b)\)  \((x,y)\cdot(a,b)=C\)

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0if we have some point \((x_0,y_0)\) such that \((x_0,y_0)\cdot(a,b)=C\) then it follows we want to find other points \((x,y)\) such that the vector from \((x_0,y_0)\) to \((x,y)\) are orthogonal to \((a,b)\), since \((xx_0,yy_0)\cdot(a,b)=0\)

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0in general, if we have \(x,x_0,n\in\mathbb{R}^n\) we have \((xx_0)\cdot n=0\) singles out a subspace of one less dimension, a *hyperplane*

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0if you want to describe a line in ndimensional euclidean space, we actually need to show that changes in our n dimensions are all proportional, giving a set of at least n1 equations in our n variables of the form: $$t=xx_0=\frac{yy_0}a=\frac{zz_0}b=\dots$$in the standard plane you commonly see this in the form \(xx_0=\frac{yy_0}b\Leftrightarrow yy_0=a(xx_0)\) in three dimensions we have $$xx_0=\frac{yy_0}a=\frac{zz_0}b$$ which takes at least two equations (necessary since a line has one degree of freedom) $$yy_0=a(xx_0)\\zz_0=b(xx_0)$$

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0you can also introduce a parameter \(t\) (essentially giving us a chart of our line in space) like i wrote above to write \(n\) parametric equations in \(n+1\) variables: $$xx_0=t\implies x=x_0+t\\\frac{yy_0}a=t\implies y=y_0+at\\\frac{zz_0}b=t\implies z=z_0+bt\\\dots$$ this can be written in more concise form by using vectors \(x,x_0,u\in\mathbb{R}^n\) so $$x(t)=x_0+tu$$

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0@Empty you were close, but you need to define \(y,z\) *both* in terms of \(x\) (or any of the three in terms of hte other two, really) to get the dimension right

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0in \(\mathbb{R}^2\) these two agree because a hyperplane is just a standard line : $$(xx_0,yy_0)\cdot (a,b)=0\\a(xx_0)+b(yy_0)=0\\yy_0=\frac{a}b(xx_0)$$

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0$$\frac{yy_0}a=\frac{xx_0}{b}$$

Jaynator495
 one year ago
Best ResponseYou've already chosen the best response.1Explosive Diarrhea ? I dunno im in a weird mood today xD

Empty
 one year ago
Best ResponseYou've already chosen the best response.1\(\color{blue}{\text{Originally Posted by}}\) @oldrin.bataku @Empty you were close, but you need to define \(y,z\) *both* in terms of \(x\) (or any of the three in terms of hte other two, really) to get the dimension right \(\color{blue}{\text{End of Quote}}\) I did, just not in such a convoluted way as you.
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