Please see the attached file for the full problem.How can I use a scalar projection to show that the distance from a point?P(x, y) to the line 3*x-4*y+5 = 0 is > abs(a*x+b*y+c)/sqrt(a^2+b^2);The trick is to draw a vector B along the line between two points. One is completely arbitrary (x0,y0) and the other is the point on the line (x2,y2) which is closest to the point (x1,y1)Then, the vector between (x1,y1) and (x2,y2) must be perpendicular to the vector between (x0,y0) and (x2,y2)
This means that their dot product must be zero.
This gives you one condition.
The other conditions come from the fact that (x0,y0) and (x2,y2) must satisfy the lines equation as well.So once you set all the conditions you will be able to express x2-x1 in terms of (x1,x2) , and (y1-y2) in terms of (x1,x2) and things are pretty straight forward from there.The files attached have the full derivations and explanations.Cheers
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