If `hati` and `hatj` are unit vectors along X-and Y-axis respectively, then what is the magnitude and direction of `hati +hatj` and `hati - hatj` ?

Options

A. 2
B. 0
C. `sqrt(2)`
D. 4

Explanation

To solve the question, we need to find the magnitude and direction of the vectors \( \hat{i} + \hat{j} \) and \( \hat{i} - \hat{j} \).

### Step 1: Finding the Magnitude of \( \hat{i} + \hat{j} \)

1. **Define the vector**: Let \( \vec{A} = \hat{i} + \hat{j} \).
2. **Calculate the magnitude**: The magnitude of a vector \( \vec{A} = a\hat{i} + b\hat{j} \) is given by the formula:
\[ |\vec{A}| = \sqrt{a^2 + b^2} \]
Here, \( a = 1 \) and \( b = 1 \).
\[ |\vec{A}| = \sqrt{1^2 + 1^2} = \sqrt{1 + 1} = \sqrt{2} \]

### Step 2: Finding the Direction of \( \hat{i} + \hat{j} \)

1. **Determine the angle**: The direction can be found using the tangent function:
\[ \tan(\theta) = \frac{b}{a} = \frac{1}{1} \]
Therefore,
\[ \theta = \tan^{-1}(1) = 45^\circ \]

### Step 3: Finding the Magnitude of \( \hat{i} - \hat{j} \)

1. **Define the vector**: Let \( \vec{B} = \hat{i} - \hat{j} \).
2. **Calculate the magnitude**: Using the same formula for magnitude:
\[ |\vec{B}| = \sqrt{a^2 + b^2} \]
Here, \( a = 1 \) and \( b = -1 \).
\[ |\vec{B}| = \sqrt{1^2 + (-1)^2} = \sqrt{1 + 1} = \sqrt{2} \]

### Step 4: Finding the Direction of \( \hat{i} - \hat{j} \)

1. **Determine the angle**: The direction can be found using the tangent function:
\[ \tan(\theta) = \frac{b}{a} = \frac{-1}{1} \]
Therefore,
\[ \theta = \tan^{-1}(-1) = -45^\circ \]

### Summary of Results

- **Magnitude of \( \hat{i} + \hat{j} \)**: \( \sqrt{2} \)
- **Direction of \( \hat{i} + \hat{j} \)**: \( 45^\circ \)
- **Magnitude of \( \hat{i} - \hat{j} \)**: \( \sqrt{2} \)
- **Direction of \( \hat{i} - \hat{j} \)**: \( -45^\circ \)