Find a sequence of plane rotations that transform x 1 1 112

Find a sequence of plane rotations that transform x = [-1, 1, 1/12]^T to a multiple of [1, 0, 0]^T.

Solution

The point x (-1,1,1/12)^T will be equal to

x\'(1,-1,-1/12)

This projection of the point in x-y plane will be inclined at 45 degrees to the x axis ans its measure will be equal to(2)^1/2

^.The final point is x\'(1,0,0)^T =(-1,0,0)

this lies on the negative x axis.

Let us assume the point x makes an angle A with respect to negative z axis

then

tan A =(2)^1/2/1/12

A =57 degrees

Thus the point is inclided at 57 degrees to negative z axis

Thus to bring it in x-y plane it has to be rotated by (90-57) degrees =33 degrees

Now the component hence got is in x y plane at an angle of 45 degrees to x axis.

This is rotated by 45 degrees to bring it on x axis

The final point in on negative x axis

Thus the point which we got till now has to be ortated by 180 degrees to reach to final orientation.

In summery, the initial point has to be rotated by 33 degrees towards the x-y plane.,then 45 degrees towards x axis and then by 180 degrees towards negative x axis to reach itas final orientation.