- #1
Gregg
- 459
- 0
[itex] \left(
\begin{array}{ccc}
0 & 1 & 0 \\
0 & 0 & 1 \\
1 & 0 & 0
\end{array}
\right) [/itex] represents a rotation.
(a) find the axis of the rotation
[itex]
\left(
\begin{array}{ccc}
0 & 1 & 0 \\
0 & 0 & 1 \\
1 & 0 & 0
\end{array}
\right)
\left(
\begin{array}{c}
x \\
y \\
z
\end{array}
\right) = \left(
\begin{array}{c}
y \\
z \\
x
\end{array}
\right)
[/itex]
[itex]
\Rightarrow y=x=z
[/itex]
(b) what is the angle of rotation
I found a perpendicular vector.
[itex]
\left(
\begin{array}{c}
1 \\
1 \\
1
\end{array}\right) \times \left(
\begin{array}{c}
-1 \\
1 \\
1
\end{array}
\right) = 0 \Rightarrow \theta = 90
[/itex]
Transform the perpendicular vector.
[itex] \left(
\begin{array}{ccc}
0 & 1 & 0 \\
0 & 0 & 1 \\
1 & 0 & 0
\end{array}
\right)\left(
\begin{array}{c}
-1 \\
1 \\
1
\end{array}
\right)
= \left(
\begin{array}{c}
1 \\
1 \\
-1
\end{array}
\right) [/itex]
Product of the perpendicular and transformed perpendicular
[itex]
\left(
\begin{array}{c}
-1 \\
1 \\
1
\end{array}
\right) \times
\left(
\begin{array}{c}
1 \\
1 \\
-1
\end{array}
\right) = -2i-2k [/itex]
this does not indicate the 120 degree rotation that i need.
[itex]
\left(
\begin{array}{c}
-1 \\
1 \\
1
\end{array}
\right)\left(
\begin{array}{c}
1 \\
1 \\
-1
\end{array}
\right) = -1 = \sqrt{3}\cos\theta \Rightarrow \theta = 109[/itex]
Is the perpendicular vector wrong? Am I trying to solve this correctly?
\begin{array}{ccc}
0 & 1 & 0 \\
0 & 0 & 1 \\
1 & 0 & 0
\end{array}
\right) [/itex] represents a rotation.
(a) find the axis of the rotation
[itex]
\left(
\begin{array}{ccc}
0 & 1 & 0 \\
0 & 0 & 1 \\
1 & 0 & 0
\end{array}
\right)
\left(
\begin{array}{c}
x \\
y \\
z
\end{array}
\right) = \left(
\begin{array}{c}
y \\
z \\
x
\end{array}
\right)
[/itex]
[itex]
\Rightarrow y=x=z
[/itex]
(b) what is the angle of rotation
I found a perpendicular vector.
[itex]
\left(
\begin{array}{c}
1 \\
1 \\
1
\end{array}\right) \times \left(
\begin{array}{c}
-1 \\
1 \\
1
\end{array}
\right) = 0 \Rightarrow \theta = 90
[/itex]
Transform the perpendicular vector.
[itex] \left(
\begin{array}{ccc}
0 & 1 & 0 \\
0 & 0 & 1 \\
1 & 0 & 0
\end{array}
\right)\left(
\begin{array}{c}
-1 \\
1 \\
1
\end{array}
\right)
= \left(
\begin{array}{c}
1 \\
1 \\
-1
\end{array}
\right) [/itex]
Product of the perpendicular and transformed perpendicular
[itex]
\left(
\begin{array}{c}
-1 \\
1 \\
1
\end{array}
\right) \times
\left(
\begin{array}{c}
1 \\
1 \\
-1
\end{array}
\right) = -2i-2k [/itex]
this does not indicate the 120 degree rotation that i need.
[itex]
\left(
\begin{array}{c}
-1 \\
1 \\
1
\end{array}
\right)\left(
\begin{array}{c}
1 \\
1 \\
-1
\end{array}
\right) = -1 = \sqrt{3}\cos\theta \Rightarrow \theta = 109[/itex]
Is the perpendicular vector wrong? Am I trying to solve this correctly?