Directed distance to a directed line

by Baba-k
Tags: directed, distance, line
Baba-k is offline
Mar13-11, 10:06 AM
P: 9

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 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 [tex]\beta[/tex]. Then [tex]\alpha = 90^{\circ} + \beta , \sin \alpha = \cos \beta , cos\alpha = -\sin \beta[/tex]. 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.

2. Relevant equations

3. 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
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