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train449
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I couldn't find a Linear Algebra section in the Homework forum. I think you can do this without cal... (I hope, but it's been a long day so you never know)
I am asked to prove that the shortest distance from a point P0(x0,y0) to a given line ax+by+c=0 is |ax0+by0+c|/sqrt(a2+b2)
D(P0, L) ; standard inner product, projection of a vector onto a subspace, orthogonal composition of a vector
Seeing as it's a linear algebra class I figured I should probably use some linear algebra methods. (Work attached, summary below)
However I'm still not sure how to deal with the constant term in the line equation.
I tried neglecting the C and having a subspace W: ax + by =0 : span (-b, a). I then drew a vector x = (x0,y0) and from these two I generated an equation for y
y = x - projWx. Then ||y|| is my D(P0,L)
However, I ran into an issue solving for ||y||. I could separate ||y-x|| which gives rise to an inequality but then I'm left with a nasty ||x||||w|| term in my absolute value bracket when I try and put all the terms over sqrt(a2+b2). I'd like that to equal c but that's wishful thinking.
Any help is greatly appreciated.
I am asked to prove that the shortest distance from a point P0(x0,y0) to a given line ax+by+c=0 is |ax0+by0+c|/sqrt(a2+b2)
Homework Equations
D(P0, L) ; standard inner product, projection of a vector onto a subspace, orthogonal composition of a vector
The Attempt at a Solution
Seeing as it's a linear algebra class I figured I should probably use some linear algebra methods. (Work attached, summary below)
However I'm still not sure how to deal with the constant term in the line equation.
I tried neglecting the C and having a subspace W: ax + by =0 : span (-b, a). I then drew a vector x = (x0,y0) and from these two I generated an equation for y
y = x - projWx. Then ||y|| is my D(P0,L)
However, I ran into an issue solving for ||y||. I could separate ||y-x|| which gives rise to an inequality but then I'm left with a nasty ||x||||w|| term in my absolute value bracket when I try and put all the terms over sqrt(a2+b2). I'd like that to equal c but that's wishful thinking.
Any help is greatly appreciated.