In $\mathbb R^3$ we write the equation of a plane as $ax+by+cz=d$. Do we have a similar equation of a line in $\mathbb R^3$? In my knowledge we don't have such equation. All we know is that given a point on a line $L$ and a vector parallel to $L$ we can derive a vector equation and a parametric equation but we don't have an equation in the form $f(x,y,z)=0$ is this correct?
cartesian equation of a line in the space
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geometry
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0Depends on what you allow $f$ to be. – 2012-06-02
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0$f$ is a map $\mathbb R^3\to \mathbb R$ – 2012-06-02
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2Then take $f(x, y, z) = (ax + by + cz - d)^2 + (ex + fy + gz - h)^2$. – 2012-06-02
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0this is a polynomial!! what about a line being an algebraic curve so it is given by two equations $f_i(x,y,z)=0$? – 2012-06-02
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0I don't understand your question. You said "map." $f$ is a map. Yes, lines in $\mathbb{R}^3$ are given by two (linear) equations. They can _also_ be given by one quadratic equation. (This is specific to $\mathbb{R}$.) – 2012-06-02
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0i'm confused. you are writing a line as the zero set of a polynomial $f(x,y,z)$ isn't this a surface, i mean an algebraic variety of dimension $2$ while a line is an algebraic variety of dimension $1$? – 2012-06-02