A vector `hati + sqrt(3) hatj` rotates about its tail through an angle 60° in clockwise direction then the new vector is

Options

A. `hati + sqrt(3)hatj`
B. `3hati - 4hatj`
C. `2hatj`
D. `2hati`

Answer

To solve the problem step by step, we will analyze the vector given and how it transforms when rotated.

### Step 1: Identify the initial vector
The initial vector is given as:
\[ \mathbf{v} = \hat{i} + \sqrt{3} \hat{j} \]
This means the vector has a component of 1 in the x-direction (i.e., along \(\hat{i}\)) and \(\sqrt{3}\) in the y-direction (i.e., along \(\hat{j}\)).

### Step 2: Plot the vector
To visualize the vector, we can plot it on a Cartesian coordinate system:
- The x-component is 1, and the y-component is \(\sqrt{3}\).
- The endpoint of the vector will be at the point (1, \(\sqrt{3}\)).

### Step 3: Calculate the angle with the x-axis
To find the angle \(\theta\) that the vector makes with the x-axis, we use the tangent function:
\[ \tan(\theta) = \frac{\text{y-component}}{\text{x-component}} = \frac{\sqrt{3}}{1} = \sqrt{3} \]
From trigonometry, we know that:
\[ \theta = 60^\circ \]

### Step 4: Rotate the vector
The problem states that the vector rotates clockwise by \(60^\circ\). Since the vector initially makes an angle of \(60^\circ\) with the x-axis, rotating it clockwise by \(60^\circ\) will align it with the x-axis.

### Step 5: Determine the new vector's direction
After the rotation, the vector will be directed along the positive x-axis. The magnitude of the vector remains the same.

### Step 6: Calculate the magnitude of the original vector
The magnitude of the original vector can be calculated as:
\[ |\mathbf{v}| = \sqrt{(1)^2 + (\sqrt{3})^2} = \sqrt{1 + 3} = \sqrt{4} = 2 \]

### Step 7: Write the new vector
Since the new direction is along the positive x-axis and the magnitude is 2, the new vector can be expressed as:
\[ \mathbf{v}_{\text{new}} = 2 \hat{i} \]

### Final Answer
Thus, the new vector after the rotation is:
\[ \mathbf{v}_{\text{new}} = 2 \hat{i} \]

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