Assertion: Angle between `hati+hatj` and `hati` is `45^(@)`.
Reason: `hati+hatj` is equally inclined to both `hati` and `hatj` and the angle between `hati` and `hatj` is `90^(@)`.

Options

A. If both Assetion and Reason are correct but Reason is the correct explanation of Assertion.
B. If both Assetion and Reason are correct but Reason is not the correct explanation of Assertion.
C. If Asserion is true but Reason is false
D. If Asserion is false but Reason is true

Explanation

To solve the question, we need to analyze both the assertion and the reason provided.

### Step 1: Understand the Assertion
The assertion states that the angle between the vectors \( \hat{i} + \hat{j} \) and \( \hat{i} \) is \( 45^\circ \).

### Step 2: Define the Vectors
Let:
- \( \mathbf{A} = \hat{i} + \hat{j} \)
- \( \mathbf{B} = \hat{i} \)

### Step 3: Use the Formula for the Angle Between Two Vectors
The angle \( \theta \) between two vectors can be found using the formula:
\[ \cos \theta = \frac{\mathbf{A} \cdot \mathbf{B}}{|\mathbf{A}| |\mathbf{B}|} \]

### Step 4: Calculate the Dot Product
First, calculate the dot product \( \mathbf{A} \cdot \mathbf{B} \):
\[ \mathbf{A} \cdot \mathbf{B} = (\hat{i} + \hat{j}) \cdot \hat{i} = \hat{i} \cdot \hat{i} + \hat{j} \cdot \hat{i} = 1 + 0 = 1 \]

### Step 5: Calculate the Magnitudes of the Vectors
Next, calculate the magnitudes:
- For \( \mathbf{A} \):
\[ |\mathbf{A}| = |\hat{i} + \hat{j}| = \sqrt{1^2 + 1^2} = \sqrt{2} \]
- For \( \mathbf{B} \):
\[ |\mathbf{B}| = |\hat{i}| = \sqrt{1^2} = 1 \]

### Step 6: Substitute into the Formula
Now substitute the values into the formula:
\[ \cos \theta = \frac{1}{\sqrt{2} \cdot 1} = \frac{1}{\sqrt{2}} \]

### Step 7: Find the Angle
To find \( \theta \), take the inverse cosine:
\[ \theta = \cos^{-1}\left(\frac{1}{\sqrt{2}}\right) = 45^\circ \]

### Conclusion for Assertion
The assertion is correct: the angle between \( \hat{i} + \hat{j} \) and \( \hat{i} \) is indeed \( 45^\circ \).

### Step 8: Analyze the Reason
The reason states that \( \hat{i} + \hat{j} \) is equally inclined to both \( \hat{i} \) and \( \hat{j} \), and the angle between \( \hat{i} \) and \( \hat{j} \) is \( 90^\circ \).

### Step 9: Visualize the Vectors
When plotted on a Cartesian plane:
- \( \hat{i} \) is along the x-axis.
- \( \hat{j} \) is along the y-axis.
- The vector \( \hat{i} + \hat{j} \) forms a 45-degree angle with both axes.

### Conclusion for Reason
The reason is also correct: \( \hat{i} + \hat{j} \) is equally inclined to both \( \hat{i} \) and \( \hat{j} \), and the angle between \( \hat{i} \) and \( \hat{j} \) is \( 90^\circ \).

### Final Conclusion
Both the assertion and the reason are correct, and the reason is a correct explanation of the assertion.

### Answer
Both the assertion and reason are correct.

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