170k views
3 votes
a particle is located on the coordinate plane at $(5,0)$. define a ''move'' for the particle as a counterclockwise rotation of $\frac{\pi}{4}$ radians about the origin followed by a translation of $10$ units in the positive $x$-direction. find the particle's position after $150$ moves.

User Cybaek
by
8.8k points

1 Answer

4 votes

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).

User Tschwab
by
7.9k points
Welcome to QAmmunity.org, where you can ask questions and receive answers from other members of our community.

9.4m questions

12.2m answers

Categories