A vector of length `l` is turned through the angle `theta` about its tail. What is the change in the position vector of its head ?

Explanation

To solve the problem of finding the change in the position vector of the head of a vector of length \( l \) that is turned through an angle \( \theta \) about its tail, we can follow these steps:

### Step 1: Define the Initial Position Vector
Let’s denote the initial position vector of the head of the vector as \( \mathbf{r} \). Since the vector has a length \( l \), we can represent it in a two-dimensional coordinate system as:
\[ \mathbf{r} = l \hat{i} \]
where \( \hat{i} \) is the unit vector along the x-axis.

### Step 2: Determine the Final Position Vector
When the vector is turned through an angle \( \theta \) about its tail, the new position vector \( \mathbf{m} \) of the head can be represented using polar coordinates. The components of the vector after rotation can be expressed as:
\[ \mathbf{m} = l \cos(\theta) \hat{i} + l \sin(\theta) \hat{j} \]
where \( \hat{j} \) is the unit vector along the y-axis.

### Step 3: Calculate the Change in Position Vector
The change in the position vector \( \mathbf{p} \) of the head can be calculated as:
\[ \mathbf{p} = \mathbf{m} - \mathbf{r} \]
Substituting the expressions for \( \mathbf{m} \) and \( \mathbf{r} \):
\[ \mathbf{p} = \left( l \cos(\theta) \hat{i} + l \sin(\theta) \hat{j} \right) - \left( l \hat{i} \right) \]
Simplifying this, we get:
\[ \mathbf{p} = l \cos(\theta) \hat{i} + l \sin(\theta) \hat{j} - l \hat{i} \]
\[ \mathbf{p} = l (\cos(\theta) - 1) \hat{i} + l \sin(\theta) \hat{j} \]

### Step 4: Final Result
Thus, the change in the position vector of the head of the vector when it is turned through an angle \( \theta \) is:
\[ \mathbf{p} = l (\cos(\theta) - 1) \hat{i} + l \sin(\theta) \hat{j} \]