If `a=2hati+5hatj and b=2hati-hatj`, then the unit vector along a+b will be

Options

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

Explanation

To find the unit vector along the vector \( \mathbf{a} + \mathbf{b} \), follow these steps:

### Step 1: Define the vectors
Given:
\[ \mathbf{a} = 2\hat{i} + 5\hat{j} \]
\[ \mathbf{b} = 2\hat{i} - \hat{j} \]

### Step 2: Calculate \( \mathbf{a} + \mathbf{b} \)
We add the vectors \( \mathbf{a} \) and \( \mathbf{b} \):
\[ \mathbf{a} + \mathbf{b} = (2\hat{i} + 5\hat{j}) + (2\hat{i} - \hat{j}) \]
Combine like terms:
\[ = (2 + 2)\hat{i} + (5 - 1)\hat{j} = 4\hat{i} + 4\hat{j} \]

### Step 3: Find the magnitude of \( \mathbf{a} + \mathbf{b} \)
The magnitude \( |\mathbf{a} + \mathbf{b}| \) is calculated as follows:
\[ |\mathbf{a} + \mathbf{b}| = \sqrt{(4)^2 + (4)^2} = \sqrt{16 + 16} = \sqrt{32} = 4\sqrt{2} \]

### Step 4: Calculate the unit vector
The unit vector \( \hat{k} \) in the direction of \( \mathbf{a} + \mathbf{b} \) is given by:
\[ \hat{k} = \frac{\mathbf{a} + \mathbf{b}}{|\mathbf{a} + \mathbf{b}|} \]
Substituting the values:
\[ \hat{k} = \frac{4\hat{i} + 4\hat{j}}{4\sqrt{2}} = \frac{4}{4\sqrt{2}}(\hat{i} + \hat{j}) = \frac{1}{\sqrt{2}}(\hat{i} + \hat{j}) \]

### Final Answer
Thus, the unit vector along \( \mathbf{a} + \mathbf{b} \) is:
\[ \hat{k} = \frac{\hat{i} + \hat{j}}{\sqrt{2}} \]

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