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

Mylo O'Moore 2021-02-06 Answered
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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Expert Answer

lobeflepnoumni
Answered 2021-02-07 Author has 99 answers
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=[cosθsinθ0sinθcosθ0001]
The vector (x, y, z) is rotated by 90 along x-axis then the vector becomes
[cos90sin900sin90cos90001][xyz]
=[010100001][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]=[151020]
By the equality of two matrices we have
{5y=155x=10z=20={y=3x=2z=20
Thus, the original vector is (-2, -3, 20)
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