Function to return a vector after it is divided by (1 + square magnitude)?

Discussion in 'Linear and Abstract Algebra' started by anonymousinquirer, Aug 8, 2023.

  1. anonymousinquirer


    Aug 8, 2023
    Likes Received:
    The square magnitude (SM) of a 3 component vector is: (xx + yy + z*z). If you divide vector A by ( 1.0 + SMA), where SMA is the square magnitude of A, is there a function that you can perform to obtain the original vector?

    Lets take original vector3 A = (Ax, Ay, Az). The final vector3 B = A/(1.0 + AxAx + AyAy + Az*Az).

    Now, is there a function of B that will return A?

    Thank you!

    I originally found that the dot product of a Vector with itself equals the vectors square magnitude. However, I also read that the inverse of dot product is impossible because of many solutions. Still, I am hoping that this case contains qualifiers that may restrict the solution set to one or two possible solution.

    I think that I worked out so far that SMA and SMB have the quadratic relationship that make SMA = -(2SMB - 1) +/- sqrt( (2SMB-1)^2 - 4(SMB)^2 )

    And I believe you could punch in this definition to satisfy the relationship A = B(1 + SMA).

    Would this mean you have to possible solutions for each component of A? Is my math correct?
    anonymousinquirer, Aug 8, 2023
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