How to change a co ordination system?

Question

I find the coordinates of given point is 210 degree (-8,-5) frm origin under normal coordination system. Now the coordination frame is tilted to 30 degree clk wise and new orgin located at 30 degree (3,3) in first quadrand of old one.The point remain unchanged in it position. Then what will be the cordinates for that point based on tilted coordination frame.

Answer 1

The first point is not at exactly 210 degrees. It is a little over 212 degrees.

The second point (3,3) is obviously at 45 degrees, not 30 degrees.

The distance between (-8, -5) and (3, 3) is
sqrt((-8 - 3)^2 + (-5 - 3)^2) = sqrt((-11)^2 + (-8)^2) = sqrt(121 + 64)
= sqrt(185)

The original angle from (3,3) to (-8, -5) is
atan2(-8 - 3, -5 - 3) = atan2(-11, -8) = 180 degrees + arctan(8/11)
counterclockwise.

The clockwise rotation subtracts thirty degrees from that, so in the final coordinate system the original (-8, -5) is sqrt(185) magnitude in the 150 degrees + arctan(8/11) direction from the new origin at the original (3,3).

Luckily, sin( arctan(8/11) ) = 8 / sqrt(8^2 + 11^2) = 8 / sqrt(185)
and cos( arctan(8/11) ) = 11 / sqrt(185).

We have the "new" x is

sqrt(185) cos(150deg. + arctan(8/11)) =
= sqrt(185) ( cos( 150 deg) cos ( arctan(8/11) ) - sin( 150 deg ) sin( arctan(8/11) ) )
= 11 cos(150 deg) - 8 sin(150 deg)
= 11 (-sqrt(3)/2) - 8 (1/2)
= -(1/2)(8 + 11 sqrt(3)) or approx. - 13.526

and the "new" y is:

sqrt(185) sin(150deg. + arctan(8/11)) =
= sqrt(185) ( sin( 150 deg ) cos ( arctan(8/11) ) + cos( 150 deg ) sin( arctan(8/11) ) )
= 11 sin( 150 deg ) + 8 cos( 150 deg )
= 11 (1/2) + 8 (- sqrt(3) / 2)
= (1/2)(11 - 8 sqrt(3)) or approx. -1.428

which makes sense, as the original slightly over 212 degrees after the rotation is slightly over 212 - 30 = 182 degrees, so both quantities are negative with the magnitude of x large and the magnitude of y small.

Check: "new x"^2 + "new y"^2 =
= ( -(1/2)(8 + 11 sqrt(3)) )^2 + ( (1/2)(11 - 8sqrt(3)) )^2
= (1/4)( 8^2 + 2*8*11 sqrt(3) + 3*11^2 + 11^2 - 2*11*8 sqrt(3) + 3*8^2)
= (1/4)( 4 (8^2 + 11^2) )
= 185 as it should be.

Check: atan2("new x", "new y") = 180 degrees + arctan("new y" / "new x")

"new y" / "new x" =
= (11 - 8sqrt(3)) / (- 8 - 11sqrt(3))
= (11 - 8sqrt(3)) (- 8 + 11sqrt(3)) / ( (-8)^2 - (11sqrt(3))^2 )
= (-88 + 121sqrt(3) + 64sqrt(3) - 88*3) / (64 - 121*3)
= (- 352 + 185sqrt(3)) / -299 or approx. 0.105587
with arctan 6.02737
and expected value arctan(8/11) - 30 or approx. 6.02737
so this checks out, too.

Changed coordinates:

( -(1/2)(8 + 11 sqrt(3)), (1/2)(11 - 8 sqrt(3)) ) or approx ( - 13.526, -1.428 )