A vector is first rotated by 90∘ along x-axis and then scaled up by 5 times is equal to (15,−10,20). What was the original vector

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lobeflepnoumni · Accepted Answer

Let us consider the vector (x, y, z).  The vector is first rotated by 90∘ along x - axis and then scaled up by 5 times.  Ratation: In homogeneous coordinates, the matrix for a rotetion about the origin through a given angle θ is B=[cos⁡θ−sin⁡θ0sin⁡θcos⁡θ0001]  The vector (x, y, z) is rotated  by 90∘ along x-axis then the vector becomes  [cos⁡90∘−sin⁡90∘0sin⁡90∘cos⁡90∘001][xyz]  =[0−10100001][xyz]  =[−yxz]  Now, the vector is scaled up by 5 times.  Scaling: In homogeneous coordinates, a scaling about the origin with scale factors c in the x-direction and c in the y-direction is computed by multiplying the vector by the matrix  A=[c000c0001]  Thus, the vector becomes  [500050001][xyz]  =[−5y5xz]  Now, given that the reduced vector is (15,−10,20). Therefore, we have  [−5y5xz]=[15−1020]  By the equality of two matrices we have  {−5y=155x=−10z=20={y=−3x=−2z=20  Thus, the original vector is (-2, -3, 20)

A vector is first rotated by displaystyle{90}^{circ} along x-axis and then scaled up by 5 times is equal to displaystyle{left({15},-{10},{20}right)}. What was the original vector

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