Which of the followingis a unit vector ?

Options

A. `hati+hatj`
B. `costheta hati-sin thetahatj`
C. `sinthetahati +2costheta hatj`
D. `(1)/(sqrt(3))(hati+ hatj)`

Explanation

To determine which of the given options is a unit vector, we need to follow these steps:

### Step 1: Understand the Definition of a Unit Vector
A unit vector is defined as a vector that has a magnitude of 1. Therefore, to identify a unit vector, we need to calculate the magnitude of each option and check if it equals 1.

### Step 2: Analyze Each Option

**Option 1: Magnitude of \( \mathbf{i} + \mathbf{j} \)**
- The magnitude of \( \mathbf{i} + \mathbf{j} \) is calculated as:
\[ |\mathbf{i} + \mathbf{j}| = \sqrt{(1)^2 + (1)^2} = \sqrt{2} \]
- Since \( \sqrt{2} \neq 1 \), this option is not a unit vector.

**Option 2: Magnitude of \( \cos \theta \mathbf{i} - \sin \theta \mathbf{j} \)**
- The magnitude is calculated as:
\[ |\cos \theta \mathbf{i} - \sin \theta \mathbf{j}| = \sqrt{(\cos \theta)^2 + (-\sin \theta)^2} = \sqrt{\cos^2 \theta + \sin^2 \theta} = \sqrt{1} = 1 \]
- Since the magnitude is 1, this option is a unit vector.

**Option 3: Magnitude of \( \sin \theta \mathbf{i} + 2 \cos \theta \mathbf{j} \)**
- The magnitude is calculated as:
\[ |\sin \theta \mathbf{i} + 2 \cos \theta \mathbf{j}| = \sqrt{(\sin \theta)^2 + (2 \cos \theta)^2} = \sqrt{\sin^2 \theta + 4 \cos^2 \theta} \]
- This can be rewritten as:
\[ = \sqrt{1 + 3 \cos^2 \theta} \]
- Since this expression does not equal 1 for all values of \( \theta \), this option is not a unit vector.

**Option 4: Magnitude of \( \frac{\mathbf{i} + \mathbf{j}}{\sqrt{3}} \)**
- The magnitude is calculated as:
\[ \left| \frac{\mathbf{i} + \mathbf{j}}{\sqrt{3}} \right| = \frac{1}{\sqrt{3}} |\mathbf{i} + \mathbf{j}| = \frac{1}{\sqrt{3}} \sqrt{2} = \frac{\sqrt{2}}{\sqrt{3}} \neq 1 \]
- Since this does not equal 1, this option is not a unit vector.

### Step 3: Conclusion
After analyzing all the options, we find that only **Option 2** has a magnitude of 1, making it the only unit vector among the choices.

### Final Answer:
**Option 2: \( \cos \theta \mathbf{i} - \sin \theta \mathbf{j} \)** is a unit vector.

---