For an Analytical Geometry course for 1st years, the book we are following went on to talk about projections after 2D geometry part(topics covered so far: coordinates, transformations, straight line, pair of straight lines and conics). There is a certain lemma I did not understand. It goes as follows:
Given a straight line,L, through origin, if $\alpha, \beta,\gamma$ are the angle L makes with the 3 axes respectively, then $l=\cos\alpha, m=\cos\beta,n=\cos\gamma$ are L's direction cosines. let $P(x,y,z)$ be any point on L and let OP have measure r.
The text then writes $x=lr,y=mr,z=rn$ which I understand to be due to theorems regarding projections on lines and all. From here, it writes: $\frac{x}{l}=\frac{y}{m}=\frac{z}{n}=r$ which is just the coordinates of a point P in terms of distance OP=r from origin and the direction ratios. Upto this i get. But then the textbook writes something I still dont understand:
let a,b,c be given proportionals to the direction cosines of a line. Then: $$\frac{\cos\alpha}{a}=\frac{\cos\beta}{b}=\frac{\cos\gamma}{c}.....(1)$$ How are we able to write this? What does it mean? After that they had written $$\frac{\cos\alpha}{a}=\frac{\cos\beta}{b}=\frac{\cos\gamma}{c}=\frac{\sqrt{(\cos^2\alpha+\cos^2\beta+\cos^2\gamma)}}{\sqrt{(a^2+b^2+c^2)}}.....(2)$$
How are we able to write (2) from (1)? I know these are all proportions, but what am I missing? I haven't seen such a property of proportions before. Is there a proof to this?
The simplest way is to verify it.
Let $\frac{cos\alpha}{a}=\frac{cos\beta}{b}=\frac{cos\gamma}{c} = k$; for some non-zero constant $k$.
Then, $\cos\alpha = ka$ and $\cos\beta = kb$ and … …
Therefore, $\dfrac {\sqrt{(cos^2\alpha+cos^2\beta+cos^2\gamma)}}{\sqrt{(a^2+b^2+c^2)}} = k = \frac{cos\alpha}{a}=\frac{cos\beta}{b}=\frac{cos\gamma}{c}$