Scalar Projection: Find Distance Point to Line

[jesuslovesu]

Homework Statement

Use the scalar projection to show that a distance from a point P(x1, y1) to the line ax + by + c = 0 is
$$\frac{ax1 + by1 + c}{\sqrt{a^2 + b^2}}$$

Homework Equations

scalar projection = $$\frac{a . b}{|a|}$$

The Attempt at a Solution

The first thing that I did was to say that b = (x1,y1). Unfortunately I'm having a difficult time coming up with a vector for the line. Honestly,, I am having a hard time visualizing how the scalar projection would yield the distance between a point and the line. Does anyone know what I should do to start?

 

[Dick]

First, points aren't vectors. Vectors are differences between points. Second, to use scalar projection the distance between a point and a line is the scalar product of a unit normal to the line with a difference vector between the point and a point on the line. Can you find a unit normal to the line?

 

[jesuslovesu]

Ok got it thanks.

 

[Dick]

I'm not sure what you are up to in getting that 'normal' vector, P1 isn't even on the line (0,-c/b) is. Start by finding a tangent vector to the line. Since y=(-1/b)x-c/b and point on the line is (x,y), if I take d/dx, I get a tangent vector of (1,-a/b), right? A normal vector is perpendicular to that, like (a,b)? Now remember after you normalize it that you want to dot it with the DIFFERENCE between (x1,y1) and a point on the line.