Find out the angle made by `(hati+hatj)` vector from X and Y axes respectively.

Explanation

To find the angles made by the vector \( \hat{i} + \hat{j} \) with the X and Y axes, we can follow these steps:

### Step 1: Understand the Vector
The vector \( \hat{i} + \hat{j} \) can be represented in a Cartesian coordinate system. Here, \( \hat{i} \) represents the unit vector along the X-axis, and \( \hat{j} \) represents the unit vector along the Y-axis. Therefore, the vector \( \hat{i} + \hat{j} \) has components:
- X-component = 1 (from \( \hat{i} \))
- Y-component = 1 (from \( \hat{j} \))

### Step 2: Draw the Vector
Draw the X and Y axes. The vector \( \hat{i} + \hat{j} \) starts from the origin (0,0) and ends at the point (1,1).

### Step 3: Identify the Angles
Let \( \theta \) be the angle made with the X-axis, and \( \phi \) be the angle made with the Y-axis.

### Step 4: Use Trigonometric Ratios
To find \( \theta \):
- We know that:
\[ \tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{1}{1} = 1 \]
- Therefore,
\[ \theta = \tan^{-1}(1) = 45^\circ \]

### Step 5: Find the Angle with Y-axis
Since the angle between the X-axis and Y-axis is \( 90^\circ \):
- We can express the relationship between \( \theta \) and \( \phi \):
\[ \theta + \phi = 90^\circ \]
- Substituting \( \theta = 45^\circ \):
\[ 45^\circ + \phi = 90^\circ \]
- Thus,
\[ \phi = 90^\circ - 45^\circ = 45^\circ \]

### Final Result
The angles made by the vector \( \hat{i} + \hat{j} \) with the X and Y axes are both \( 45^\circ \).