Ways to write a line
- slope/intercept form
- point/slope form
- point/vector form
- hyperplane form

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slope/intercept
$y=mx+b$
notation: a stupid combo of $y=mx+n$ and $y=ax+b$
- unique form
- doesn't work for vertical lines
- good for comparing lines
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point/slope
$y-b=m(x-a)$
- not unique (different points)
- natural for tangent lines to functions:
$y-f(a)=f'(a)(x-a)$
...b/c interested -near the point-
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point/vector
$\ell(t) = (a,b) + t (m,n)$
- not unique (different points -and- vectors)
- natural for tangent lines to parametric curves
- generalizes easily to more dimensions
- and indeed to tangent planes!
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hyperplane
$mx + ny = a$
in 2D, hyperplanes are lines;
this is the ``set of points'' form,
and generalizes to covectors/hyperplanes in more dimensions
...also leads naturally to projective space
