Draw the two pairs of cartesian (rectangular) axes having the same origin (0,0), such that (u,v) is rotated 45 degees from (x,y).

Mark a random point (x,0) on the positive x-axis. Draw the vertical line x = x, or project the (x,0) on the (u,v) axes.

What are the (u,v) coordinates of this line x = x?

u = x*sqrt(2)

v = -x*sqrt(2)

Or, x = (u/sqrt(2),0) +(0,-v/sqrt(2))

So,

x = u/sqrt(2) -v/sqrt(2)

x = (1/sqrt(2))*(u-v)

x = (1/2)sqrt(2) *(u-v) ---------------***

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For the y on the (u,v) axes.

Mark a random point (0,y) on the positive y-axis. Then project that onto the (u,v) axes.

u = y*sqrt(2)

v = y*sqrt(2)

Or, y = (u/sqrt(2),0) +(0,v/sqrt(2))

So, y = u/sqrt(2) +v/sqrt(2)

y = (1/2)sqrt(2) *(u+v) ---------------**