Given vec(u) =((u_1),(u_2)), if it is rotated sixty degrees anti-clockwise prove that the resulting vector will be vec(v) =1/2((u_1−sqrt(3)u_2),(sqrt(3)u_1+u_2))

Gauge Roach

Gauge Roach

Open question

2022-08-13

Given u = ( u 1 u 2 ) , if it is rotated sixty degrees anti-clockwise prove that the resulting vector will be v = 1 2 ( u 1 3 u 2 3 u 1 + u 2 )
I think there are better methods out there but one idea I had was to use cos 60 = 1 2 = u v | u | | v |
and solve for v but obviously this seems impossible to me with the | v | and I see no where to force a 3 to appear.
Is this true? Or if this method works please share. I have other ideas which probably work but this one also came to mind but with no fruitition

Answer & Explanation

Royce Morrison

Royce Morrison

Beginner2022-08-14Added 12 answers

Your method actually could work but there is an easier approach involving the Rotation matrix which is defined as
R θ = ( cos θ sin θ sin θ cos θ )
it can be easily proven that
R θ ( cos x sin x ) = ( cos ( x + θ ) sin ( x + θ ) )
Let ( u 1 u 2 ) = r ( cos x sin x )
since rotating sixty degrees anti-clockwise will result same as rotating three hundred clockwise let θ = 300
R θ ( u 1 u 2 ) = r R θ ( cos x sin x ) = r ( cos ( x + 300 ) sin ( x + 300 ) )
r ( cos ( x + 300 ) sin ( x + 300 ) ) = r ( cos ( x ) cos 300 sin x sin 300 sin x cos 300 + cos x sin 300 ) = ( r 2 cos x r 3 sin x 2 r 2 sin x + r 3 2 cos x ) = 1 2 ( u 1 3 u 2 3 u 1 + u 2 )

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