Answer:after 150 moves, the particle will be located at the coordinates (10, 10).
Explanation:
To find the particle's position after 150 moves, we can break down each move into two steps: a counterclockwise rotation of π/4 radians followed by a translation of 10 units in the positive x-direction. We can repeat these two steps 150 times.
Step 1: Counterclockwise Rotation of π/4 Radians
A counterclockwise rotation of π/4 radians can be represented by the transformation matrix:
\cos(\theta) & -\sin(\theta) \\
\sin(\theta) & \cos(\theta)
\end{bmatrix}\]
In this case, θ = π/4:
\[R(\frac{\pi}{4}) = \begin{bmatrix}
\cos(\frac{\pi}{4}) & -\sin(\frac{\pi}{4}) \\
\sin(\frac{\pi}{4}) & \cos(\frac{\pi}{4})
\end{bmatrix} = \begin{bmatrix}
\frac{\sqrt{2}}{2} & -\frac{\sqrt{2}}{2} \\
\frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2}
\end{bmatrix}\]
Step 2: Translation of 10 Units in the Positive x-Direction
A translation in the positive x-direction can be represented by adding 10 to the x-coordinate:
\[T(10) = \begin{bmatrix}
1 & 0 \\
0 & 1
\end{bmatrix}\begin{bmatrix}
10 \\
0
\end{bmatrix} = \begin{bmatrix}
10 \\
0
\end{bmatrix}\]
Now, let's apply these two steps (rotation and translation) 150 times to the initial position (5, 0).
Starting position: (5, 0)
After one move: \[R(\frac{\pi}{4}) \cdot T(10) \cdot (5, 0) = \begin{bmatrix}
\frac{\sqrt{2}}{2} & -\frac{\sqrt{2}}{2} \\
\frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2}
\end{bmatrix} \cdot \begin{bmatrix}
10 \\
0
\end{bmatrix} = \begin{bmatrix}
5\sqrt{2} \\
5\sqrt{2}
\end{bmatrix}\]
After two moves: \[R(\frac{\pi}{4}) \cdot T(10) \cdot \begin{bmatrix}
5\sqrt{2} \\
5\sqrt{2}
\end{bmatrix} = \begin{bmatrix}
\frac{\sqrt{2}}{2} & -\frac{\sqrt{2}}{2} \\
\frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2}
\end{bmatrix} \cdot \begin{bmatrix}
10 \\
0
\end{bmatrix} = \begin{bmatrix}
10 \\
10
\end{bmatrix}\]
Now, you can continue this process for a total of 150 moves. After 150 moves, the particle's position will be:
\[\begin{bmatrix}
10 \\
10
\end{bmatrix} = (10, 10)\]
So, after 150 moves, the particle will be located at the coordinates (10, 10).