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3x3 Matrix rotation question?



The matrix:
| 0 1 0 |
| 0 0 1 |
| 1 0 0 |
represents a transformation




a) Find the equation of the axis of this rotation
b) What is the angle of the rotation? (answer = 120)

so I saw that for a) it was just x=y=z but i'm a bit stuck on b) as i don't really know how i'd go about finding the angle?
(edited 7 years ago)
Original post by xianlong
just visualise it in ur head?


i did? but when i visualise it i think that the angle must be 90?
Original post by ithinkitslily






The matrix:
| 0 1 0 |
| 0 0 1 |
| 1 0 0 |
represents a transformation







a) Find the equation of the axis of this rotation
b) What is the angle of the rotation? (answer = 120)

so I saw that for a) it was just x=y=z but i'm a bit stuck on b) as i don't really know how i'd go about finding the angle?


a) To find the vector pointing along the axis of rotation, note that this vector is left unchanged by the rotation i.e. you must solve the eigenvalue equation Rv=v\bold{R} \bold{v} = \bold{v}

b) To find the angle of rotation, note that a vector u\bold{u} in a plane at right angles to v\bold{v} is rotated by the angle of rotation into u′\bold{u}' say. Then you can find the angle between these two vectors via the dot product.

Take the orthogonal plane through the origin for ease of working. That is given by v⋅(xi+yj+zk)=0\bold{v} \cdot (x\bold{i} +y\bold{j} + z\bold{k})=0

Or alternatively google the trace of a rotation matrix.
Original post by ithinkitslily
i did? but when i visualise it i think that the angle must be 90?

To do this by geometrical intuition, note that the rotation matrix takes x→z→yx \to z \to y. This is a rotation of θ\theta, say, about the axis of rotation.

How many such rotations return the axes to the original configuration?

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