Determine the rotation matrix that rotates 3-vectors through an angle of 300 in the plane x1 + x2 + x3 = 0.

Question

The subject is ANALYTICAL GEOMETRY AND VECTOR ANALYSIS

Answer 1

Let's assume this is 300° counterclockwise as viewed from above (observer is on the positive octant, e.g. at point P(1,1,1)).

 

To do this, we need a convenient basis. The best choice is an orthonormal basis with two vectors on the plane. The basis should be right-handed (positive determinant).

 

First pick an orthogonal basis: v₁,v₂ on the plane, v₃ orthogonal to it.

 

v₁ = ⟨1,1,-2⟩

v₂ = ⟨-1,1,0⟩

v₃ = ⟨1,1,1⟩

 

Normalize them.

 

u₁ = ⟨1/√6,1/√6,-2/√6⟩
u₂ = ⟨-1/√2,1/√2,0⟩
u₃ = ⟨1/√3,1/√3,1/√3⟩

 

You can verify that this is a right handed basis as det[u₁ u₂ u₃] = 1.

 

Let S be a matrix that maps the standard basis to this one.

 

S =

 

⌈ 1/√6   -1/√2    1/√3⌉

| 1/√6    1/√2    1/√3|

⌊-2/√6      0       1/√3⌋

 

Since S is orthogonal, it's inverse is just it's transpose.

 

S^-1 =

 

⌈  1/√6    1/√6   -2/√6⌉
|-1/√2    1/√2       0   |
⌊  1/√3    1/√3    1/√3⌋

 

A rotation about the x₃-axis by 300° is.

 

R=

 

⌈ cos 300°    -sin 300°    0 ⌉
| sin 300°     cos 300°    0 | =
⌊      0                0            1 ⌋

 

⌈    1/2    √3/2    0 ⌉
| -√3/2    1/2     0 |
⌊     0        0        1 ⌋

 

The solution to your problem is then

 

A = SIS^-1 =

 

⌈  2/3    2/3   -1/3 ⌉
|-1/3    2/3     2/3 |
⌊  2/3  -1/3     2/3 ⌋