Find the unit vector in the direction of the sum of the vectors :
`vec(a) = 2hat(i)-hat(j)+2hat(k)` and `vec(b)=-hat(i)+hat(j)+3hat(k)`.

Answer

To find the unit vector in the direction of the sum of the vectors \(\vec{a} = 2\hat{i} - \hat{j} + 2\hat{k}\) and \(\vec{b} = -\hat{i} + \hat{j} + 3\hat{k}\), we will follow these steps:

### Step 1: Find the sum of the vectors \(\vec{a}\) and \(\vec{b}\).

\[ \vec{c} = \vec{a} + \vec{b} \]

Substituting the values of \(\vec{a}\) and \(\vec{b}\):

\[ \vec{c} = (2\hat{i} - \hat{j} + 2\hat{k}) + (-\hat{i} + \hat{j} + 3\hat{k}) \]

### Step 2: Combine the components of the vectors.

Now, we will combine the corresponding components:

- For \(\hat{i}\) components: \(2 - 1 = 1\)
- For \(\hat{j}\) components: \(-1 + 1 = 0\)
- For \(\hat{k}\) components: \(2 + 3 = 5\)

Thus, we have:

\[ \vec{c} = 1\hat{i} + 0\hat{j} + 5\hat{k} = \hat{i} + 5\hat{k} \]

### Step 3: Calculate the magnitude of vector \(\vec{c}\).

The magnitude of \(\vec{c}\) is given by:

\[ |\vec{c}| = \sqrt{(1)^2 + (0)^2 + (5)^2} \]

Calculating this:

\[ |\vec{c}| = \sqrt{1 + 0 + 25} = \sqrt{26} \]

### Step 4: Find the unit vector in the direction of \(\vec{c}\).

The unit vector \(\hat{u}\) in the direction of \(\vec{c}\) is given by:

\[ \hat{u} = \frac{\vec{c}}{|\vec{c}|} \]

Substituting the values:

\[ \hat{u} = \frac{\hat{i} + 5\hat{k}}{\sqrt{26}} \]

### Final Answer:

Thus, the unit vector in the direction of the sum of the vectors is:

\[ \hat{u} = \frac{1}{\sqrt{26}}\hat{i} + \frac{5}{\sqrt{26}}\hat{k} \]