The unit vector along `hat(i)+hat(j)` is

Options

A. `hat(k)`
B. `hat(i)+hat(j)`
C. `(hat(i)+hat(j))/(sqrt(2))`
D. `(hat(i)+hat(j))/(2)`

Explanation

To find the unit vector along the vector \(\hat{i} + \hat{j}\), we can follow these steps:

### Step 1: Identify the vector
The given vector is \(\hat{i} + \hat{j}\).

### Step 2: Calculate the magnitude of the vector
The magnitude of a vector \(\vec{A} = a\hat{i} + b\hat{j}\) is given by the formula:
\[ |\vec{A}| = \sqrt{a^2 + b^2} \]
In our case, \(a = 1\) and \(b = 1\):
\[ |\hat{i} + \hat{j}| = \sqrt{1^2 + 1^2} = \sqrt{1 + 1} = \sqrt{2} \]

### Step 3: Find the unit vector
The unit vector \(\hat{u}\) in the direction of vector \(\vec{A}\) is given by:
\[ \hat{u} = \frac{\vec{A}}{|\vec{A}|} \]
Substituting \(\vec{A} = \hat{i} + \hat{j}\) and \(|\vec{A}| = \sqrt{2}\):
\[ \hat{u} = \frac{\hat{i} + \hat{j}}{\sqrt{2}} \]

### Final Answer
Thus, the unit vector along \(\hat{i} + \hat{j}\) is:
\[ \hat{u} = \frac{1}{\sqrt{2}} \hat{i} + \frac{1}{\sqrt{2}} \hat{j} \]

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