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I've reached a question in a maths text book I'm reading that I'm having some trouble with. Any help/suggestion would be greatly appreciated. The question it self is short but there are a couple of paragraphs which set it up that I've copied directly out of the book.

1. The problem statement, all variables and given/known data

We distinguish a positive and a negative side of a directed linelas follows. LetPbe any point not onl, and letQbe the foot of the perpendicular tolthroughP. ThenPis on the positive or negative side oflaccording as the angle fromlto the directed lineQPis 90 or -90 degrees.

We shall now determine the equation of a directed linel. We draw through the orignOa linemperpendicular tol, and directmso that the angle from it tolis 90 degrees. The angle from the directedx-axis tomwill be called [tex]\beta[/tex]. Then [tex]\alpha = 90^{\circ} + \beta , \sin \alpha = \cos \beta , cos\alpha = -\sin \beta[/tex]. LetRwith coordinates x_{1}, y_{1}be the point wheremmeetsl. We shall denote bydthe directed distanceORon directedm.

Show thatdis positive if and only ifOlies on the negative side ofl.

2. Relevant equations

3. The attempt at a solution

The question states thatdshould be positive if and only ifOis on the negative side ofl. However I've attached a simple sketch where (i think)dis positive whenOis on the positive side (the black line being l and the green one being d and m). I'm not sure what I've done wrong to get this result, any pointers would be great :)

thanks alot

babak

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# Directed distance to a directed line

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