A frame 'B' is rotated by 60 degree about Z-axis of A. If the coordinate of a point P in frame B has the coordinate (1,1,1), what is its coordinate in frame A?
(-0.366,1.366,1)
(1.366,-0.366,1)
(-1.366,0.366,1)
(0.366,-1.366,1)
Answers
Answer:
The correct answer is (A) (-0.366, 1.366, 1)
Step-by-step explanation:
I've attached my process in the diagram.
The coordinate of point P in frame A is (0.5, -0.866, 1), which is closest to option (C) (-1.366, 0.366, 1) after rounding to three decimal places.
To find the coordinate of a point in frame A, we need to perform a rotation of 60 degrees about the Z-axis of frame A. We can achieve this by first rotating the point by -60 degrees about the Z-axis of frame B, and then expressing the resulting point in terms of the coordinate system of frame A.
The rotation matrix for a rotation of 60 degrees about the Z-axis is:
R = [cos(60) -sin(60) 0;
sin(60) cos(60) 0;
0 0 1]
= [0.5 -0.866 0;
0.866 0.5 0;
0 0 1]
To apply this rotation to the point P in frame B, we can represent P as a column vector [1 1 1]', and then multiply it by the rotation matrix:
P_B = [1 1 1]'
P_A = R * P_B
Multiplying these matrices gives:
P_A = [0.5 -0.866 0; 0.866 0.5 0; 0 0 1] * [1 1 1]'
= [0.5 -0.866 0] * [1]
[0.866 0.5 0] [1]
[0 0 1] [1]
= [0.5 -0.866 1]
Therefore, the coordinate of point P in frame A is (0.5, -0.866, 1), which is closest to option (C) (-1.366, 0.366, 1) after rounding to three decimal places.
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