How Can I Visualize a $\pi/2$ Rotation About $(1,1,0)^t$ in $\mathbb{R}^3$?

[kalish1]

I am trying to visualize the following rotation of $\mathbb R^3$, but it is very difficult. I want to get the answer by intuition, and not by using the Rodrigues rotation formula or conjugation of matrices, etc.

Help please.

**Problem statement:** Determine the matrix that represents the following rotation of $\mathbb R^3$: an angle of $\pi/2$ about the fixed axis containing the vector $(1,1,0)^t$

Here is what I have tried in my diagram:

![Coordinate axes][1]

Should I find a 3x3 rotation matrix $A$ such that $A(1,1,0)^t=(-1,1,0)^t$?

[1]:
View attachment 1648

 

[I like Serena]

Should I find a 3x3 rotation matrix $A$ such that $A(1,1,0)^t=(-1,1,0)^t$?

The axis is invariant, so that should be $A(1,1,0)^t=(1,1,0)^t$ instead.

You already found a perpendicular vector (-1,1,0).
You need a 3rd vector that is perpendicular to both the axis and this one, completing an orthogonal set.
And more specifically, a vector that "turns" in the right direction.
You can find it by calculating the cross product of (1,1,0) and (-1,1,0), yielding (0,0,2).

That gives you the set of equations (note that the vectors must have equal lengths):
$$A(1,1,0)^t=(1,1,0)^t$$
$$A(-1,1,0)^t=(0,0,\sqrt 2)^t$$
$$A(0,0,\sqrt 2)^t=(1,-1,0)^t$$

Or in matrix form:
$$A \begin{bmatrix}
1 & -1 & 0 \\
1 & 1 & 0 \\
0 & 0 & \sqrt 2
\end{bmatrix} = \begin{bmatrix}
1 & 0 & 1 \\
1 & 0 & -1 \\
0 & \sqrt 2 & 0
\end{bmatrix}$$
Solve for A.
This is easiest to do by performing matrix operations on the right that execute a gaussian elimination, ending up with an identity matrix on the left and the actual A matrix on the right.
That is, subtract multiples of one column from another, both left and right.
And swap columns, both left and right.

 

[Evgeny.Makarov]

Here is a picture.

It is done using TikZ package 3dplot.

 

[I like Serena]

Perhaps noteworthy for visualization is that (-1,1,0) is rotated onto the positive z-axis,
which is in turn rotated to (1,-1,0).

 

[blue_raver22]

https://i.stack.imgur.com/WnE8K.png

I understand the difficulty of trying to visualize a rotation in three-dimensional space without the aid of mathematical formulas. However, it is important to note that intuition alone may not always provide an accurate solution.

In this case, your approach of finding a rotation matrix $A$ such that $A(1,1,0)^t=(-1,1,0)^t$ is a good start. This will indeed give you a rotation of $\pi/2$ about the fixed axis containing the vector $(1,1,0)^t$. However, this approach may not always work for more complex rotations.

It is also important to mention that there are other methods for visualizing rotations in three-dimensional space, such as using quaternions or Euler angles. These methods may be more intuitive for some individuals.

In general, while intuition can be a helpful tool in problem-solving, it is important to also use mathematical formulas and techniques to ensure accuracy and correctness in your solutions. I would recommend exploring different methods and practicing with different examples to improve your ability to visualize rotations in three-dimensional space.