"Greg Egan - Foundations 1 - Special Relativity" - читать интересную книгу автора (Egan Greg)

x2 + y2
= (3)


Eqn (4) states that the cosine of the angle between two vectors (x,y) and (u,w)
is equal to the metric function applied to the two vectors, divided by both their lengths.
Eqn (4) will take a bit of work to prove, but in doing so we'll solve the whole problem of
rotations in space.


cos B = g[(x,y),(u,w)] / (|(x,y)||(u,w)|) (4)


where B is the angle between (x,y) and (u,w).




If you want to know the x-coordinate of a point like P in Figure 3, you draw a
line through P at right angles to the x-axis, and see where it hits the axis. In the process,
the vector OP is shown to be the sum of two vectors: OQ, which is parallel to the x-
axis, and QP, which is perpendicular to it.
The same thing can be done with any other vector in place of the x-axis. If a line
from P to OG meets OG at a right angle, at point S, then OS is called the projection of
 Egan: "Foundations 1"/p.7


OP onto OG. And again, OP is shown to be the sum of two vectors: OS, which is
parallel to OG, and SP, which is perpendicular.
How long is OS? If the angle between OP and OG is B:


OS = OP cos B
= |(x,y)| cos B (5)


What if we don't know B? Suppose all we know are the coordinates of P, (x,y),
and the angle A that OG makes with the x-axis. Projecting OS onto the coordinate axes
to make OT and OU, and projecting OT back onto OS to make OV:


OS = OV + VS
= OT cos A + OU sin A
= g[(OT, OU),(cos A, sin A)]


We don't know (OT, OU), but we do know that it's the part of the vector (x,y)
parallel to OG, when (x,y) is written as a sum of parallel and perpendicular components: