Directed distance to a directed line

[Baba-k]

Hi,

I've reached a question in a maths textbook 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.

Homework Statement

We distinguish a positive and a negative side of a directed line l as follows. Let P be any point not on l, and let Q be the foot of the perpendicular to l through P. Then P is on the positive or negative side of l according as the angle from l to the directed line QP is 90 or -90 degrees.
We shall now determine the equation of a directed line l. We draw through the orign O a line m perpendicular to l, and direct m so that the angle from it to l is 90 degrees. The angle from the directed x-axis to m will be called $$\beta$$. Then $$\alpha = 90^{\circ} + \beta , \sin \alpha = \cos \beta , cos\alpha = -\sin \beta$$. Let R with coordinates x₁, y₁ be the point where m meets l. We shall denote by d the directed distance OR on directed m.

Show that d is positive if and only if O lies on the negative side of l.

Homework Equations

The Attempt at a Solution

The question states that d should be positive if and only if O is on the negative side of l. However I've attached a simple sketch where (i think) d is positive when O is 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

 

[LightHero]

It looks like you have drawn the diagram incorrectly. The angle \beta should be pointing in the opposite direction, so that it is on the same side of l as O. Then, when O is on the negative side of l, d will be positive and when O is on the positive side of l, d will be negative.