Question

What are the position vectors of the tip of the hour hand and the tip of the minute hand for a clock with a minute hand length of 0.49 m and an hour hand length of 0.26 m at 12:00, 3:00, 6:00 and 9:00?

Answer 1

To determine the position vectors of the tips of the hour hand and minute hand of a clock at specified times, we can use polar coordinates, where the angle is measured counterclockwise from the positive x-axis. The position vector \(\mathbf{r}\) of the tip of a hand can be expressed as:

\[
\mathbf{r} = r \begin{pmatrix}
\cos(\theta) \\
\sin(\theta)
\end{pmatrix}
\]

where \(r\) is the length of the hand and \(\theta\) is the angle of the hand at that time.

### 1. At 12:00
- **Minute Hand (length = 0.49 m):**
- Angle \(\theta = 0\) radians (pointing to the right)
- Position vector:
\[
\mathbf{r}_{\text{minute}} = 0.49 \begin{pmatrix}
\cos(0) \\
\sin(0)
\end{pmatrix} = 0.49 \begin{pmatrix}
1 \\
0
\end{pmatrix} = \begin{pmatrix}
0.49 \\
0
\end{pmatrix}
\]

- **Hour Hand (length = 0.26 m):**
- Angle \(\theta = 0\) radians (also pointing to the right)
- Position vector:
\[
\mathbf{r}_{\text{hour}} = 0.26 \begin{pmatrix}
\cos(0) \\
\sin(0)
\end{pmatrix} = 0.26 \begin{pmatrix}
1 \\
0
\end{pmatrix} = \begin{pmatrix}
0.26 \\
0
\end{pmatrix}
\]

### 2. At 3:00
- **Minute Hand:**
- Angle \(\theta = \frac{\pi}{2}\) radians (pointing upwards)
- Position vector:
\[
\mathbf{r}_{\text{minute}} = 0.49 \begin{pmatrix}
\cos\left(\frac{\pi}{2}\right) \\
\sin\left(\frac{\pi}{2}\right)
\end{pmatrix} = 0.49 \begin{pmatrix}
0 \\
1
\end{pmatrix} = \begin{pmatrix}
0 \\
0.49
\end{pmatrix}
\]

- **Hour Hand:**
- Angle \(\theta = \frac{\pi}{6}\) radians (15 degrees)
- Position vector:
\[
\mathbf{r}_{\text{hour}} = 0.26 \begin{pmatrix}
\cos\left(\frac{\pi}{6}\right) \\
\sin\left(\frac{\pi}{6}\right)
\end{pmatrix} = 0.26 \begin{pmatrix}
\frac{\sqrt{3}}{2} \\
\frac{1}{2}
\end{pmatrix} = \begin{pmatrix}
0.26 \cdot \frac{\sqrt{3}}{2} \\
0.26 \cdot \frac{1}{2}
\end{pmatrix} = \begin{pmatrix}
0.225 \\
0.13
\end{pmatrix}
\]

### 3. At 6:00
- **Minute Hand:**
- Angle \(\theta = \pi\) radians (pointing to the left)
- Position vector:
\[
\mathbf{r}_{\text{minute}} = 0.49 \begin{pmatrix}
\cos(\pi) \\
\sin(\pi)
\end{pmatrix} = 0.49 \begin{pmatrix}
-1 \\
0
\end{pmatrix} = \begin{pmatrix}
-0.49 \\
0
\end{pmatrix}
\]

- **Hour Hand:**
- Angle \(\theta = \frac{\pi}{3}\) radians (60 degrees)
- Position vector:
\[
\mathbf{r}_{\text{hour}} = 0.26 \begin{pmatrix}
\cos\left(\frac{\pi}{3}\right) \\
\sin\left(\frac{\pi}{3}\right)
\end{pmatrix} = 0.26 \begin{pmatrix}
\frac{1}{2} \\
\frac{\sqrt{3}}{2}
\end{pmatrix} = \begin{pmatrix}
0.13 \\
0.225
\end{pmatrix}
\]

### 4. At 9:00
- **Minute Hand:**
- Angle \(\theta = \frac{3\pi}{2}\) radians (pointing downwards)
- Position vector:
\[
\mathbf{r}_{\text{minute}} = 0.49 \begin{pmatrix}
\cos\left(\frac{3\pi}{2}\right) \\
\sin\left(\frac{3\pi}{2}\right)
\end{pmatrix} = 0.49 \begin{pmatrix}
0 \\
-1
\end{pmatrix} = \begin{pmatrix}
0 \\
-0.49
\end{pmatrix}
\]

- **Hour Hand:**
- Angle \(\theta = \frac{5\pi}{6}\) radians (150 degrees)
- Position vector:
\[
\mathbf{r}_{\text{hour}} = 0.26 \begin{pmatrix}
\cos\left(\frac{5\pi}{6}\right) \\
\sin\left(\frac{5\pi}{6}\right)
\end{pmatrix} = 0.26 \begin{pmatrix}
-\frac{\sqrt{3}}{2} \\
\frac{1}{2}
\end{pmatrix} = \begin{pmatrix}
-0.225 \\
0.13
\end{pmatrix}
\]

### Summary:
Position vectors of the tips of the hour hand and minute hand at different times are:

- **At 12:00:**
- Minute hand: \(\begin{pmatrix} 0.49 \\ 0 \end{pmatrix}\)
- Hour hand: \(\begin{pmatrix} 0.26 \\ 0 \end{pmatrix}\)

- **At 3:00:**
- Minute hand: \(\begin{pmatrix} 0 \\ 0.49 \end{pmatrix}\)
- Hour hand: \(\begin{pmatrix} 0.225 \\ 0.13 \end{pmatrix}\)

- **At 6:00:**
- Minute hand: \(\begin{pmatrix} -0.49 \\ 0 \end{pmatrix}\)
- Hour hand: \(\begin{pmatrix} 0.13 \\ 0.225 \end{pmatrix}\)

- **At 9:00:**
- Minute hand: \(\begin{pmatrix} 0 \\ -0.49 \end{pmatrix}\)
- Hour hand: \(\begin{pmatrix} -0.225 \\ 0.13 \end{pmatrix}\)