Find the unit vector in the direction of the vector :
`vec(b)=2hat(i)+hat(j)+2hat(k)`

Answer

To find the unit vector in the direction of the vector **b** given by:

\[ \vec{b} = 2\hat{i} + \hat{j} + 2\hat{k} \]

we will follow these steps:

### Step 1: Identify the vector
The vector **b** is given as:

\[ \vec{b} = 2\hat{i} + \hat{j} + 2\hat{k} \]

### Step 2: Calculate the magnitude of the vector
The magnitude of a vector \(\vec{v} = a\hat{i} + b\hat{j} + c\hat{k}\) is calculated using the formula:

\[ |\vec{v}| = \sqrt{a^2 + b^2 + c^2} \]

For our vector **b**:

- \(a = 2\)
- \(b = 1\)
- \(c = 2\)

Thus, the magnitude of **b** is:

\[ |\vec{b}| = \sqrt{2^2 + 1^2 + 2^2} = \sqrt{4 + 1 + 4} = \sqrt{9} = 3 \]

### Step 3: Calculate the unit vector
The unit vector in the direction of **b** is given by:

\[ \hat{b} = \frac{\vec{b}}{|\vec{b}|} \]

Substituting the values we have:

\[ \hat{b} = \frac{2\hat{i} + \hat{j} + 2\hat{k}}{3} \]

### Step 4: Write the final answer
Thus, the unit vector in the direction of the vector **b** is:

\[ \hat{b} = \frac{2}{3}\hat{i} + \frac{1}{3}\hat{j} + \frac{2}{3}\hat{k} \]

### Summary of the Solution
The unit vector in the direction of the vector \( \vec{b} = 2\hat{i} + \hat{j} + 2\hat{k} \) is:

\[ \hat{b} = \frac{2}{3}\hat{i} + \frac{1}{3}\hat{j} + \frac{2}{3}\hat{k} \]

---