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Math Help - Proving the counterclockwise rotation about the z-axis

  1. #1
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    Proving the counterclockwise rotation about the z-axis

    Hi,

    I'm having trouble proving the fact that the counterclockwise rotation about the positive z-axis through an angle θ becomes

    w_1= xcosθ-ysinθ
    w_2= xsinθ+ycosθ
    w_3= z

    when X=(x,y,z) is rotated counterclockwise to obtain w=(w_1,w_2,w_3)
    I understand how this would hold when the diameter of the base of the cone(in this case =2r) is set along the x-axis, and the angle between the x-axis and X is defined by φ.

    Explanation:
    As x=rcosφ, y=rsinφ,
    x=cos(θ+φ), y=sin(θ+φ).
    From the addition theorem of trigonometric functions,

    w_1= xcosθ-ysinθ
    w_2= xsinθ+ycosθ
    w_3= z

    But wouldn't w_1 and w_2 be defined differently if the diameter of the base of the cone is on the y-axis? Wouldn't x= rsinθ,y=rcosθ?
    Please help!
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  2. #2
    MHF Contributor

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    Re: Proving the counterclockwise rotation about the z-axis

    suppose we take our "0 angle" to be the (positive) y-axis (i think this is what you are saying).

    in this case, if φ is the angle between the point (x,y) and the y-axis:

    x = rcos(φ + π/2) = rcos(π/2 - (-φ)) = rsin(-φ) = -rsin(φ)
    y = rsin(φ + π/2) = rsin(π/2 - (-φ)) = rcos(-φ) = rcos(φ)

    after rotating through an angle of θ, our new coordinates are:

    w1 = -rsin(φ+θ)
    w2 = rcos(φ+θ)
    w3 = z (the z-axis is unaffected by a rotation in the xy-plane).

    using the SAME trig identities as before, we have:

    w1 = xsin(θ) - ycos(θ)
    w2 = ycos(θ) + xsin(θ) = xsin(θ) + ycos(θ)

    so, no...it doesn't matter "which axis" we count our "rotation from".

    if you think about it, the formula for a cone:

    z2 = x2+y2, is the same even if we "switch the x's and y's".

    in fact, you could start measuring "the original angle of the point (x,y)" from ANY line in the xy-plane passing through the origin, and the formula for the w's would STILL come out the same, but the algebra gets more complicated (because the formula for x and y would be defined in terms of an angle from a line which is itself at some angle from the x-axis).

    put another way, ALL diameters of a circle (and hence of a "slice of a cone parallel to the xy-plane") work just as well, so why not pick a "convenient" one.
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