Find the unit vector in the direction of the vector :
`vec(a)=3hat(i)+2hat(j)+6hat(k)`

Answer

To find the unit vector in the direction of the vector \(\vec{a} = 3\hat{i} + 2\hat{j} + 6\hat{k}\), we will follow these steps:

### Step 1: Write down the vector
The given vector is:
\[ \vec{a} = 3\hat{i} + 2\hat{j} + 6\hat{k} \]

### Step 2: Calculate the magnitude of the vector
The magnitude of a vector \(\vec{a} = a_1\hat{i} + a_2\hat{j} + a_3\hat{k}\) is given by the formula:
\[ |\vec{a}| = \sqrt{a_1^2 + a_2^2 + a_3^2} \]
For our vector:
- \(a_1 = 3\)
- \(a_2 = 2\)
- \(a_3 = 6\)

Now, substituting these values:
\[ |\vec{a}| = \sqrt{3^2 + 2^2 + 6^2} = \sqrt{9 + 4 + 36} = \sqrt{49} = 7 \]

### Step 3: Find the unit vector
The unit vector \(\hat{a}\) in the direction of \(\vec{a}\) is given by:
\[ \hat{a} = \frac{\vec{a}}{|\vec{a}|} \]
Substituting the values we have:
\[ \hat{a} = \frac{3\hat{i} + 2\hat{j} + 6\hat{k}}{7} \]

### Step 4: Write the final answer
Thus, the unit vector in the direction of \(\vec{a}\) is:
\[ \hat{a} = \frac{3}{7}\hat{i} + \frac{2}{7}\hat{j} + \frac{6}{7}\hat{k} \]

### Summary
The unit vector in the direction of the vector \(\vec{a} = 3\hat{i} + 2\hat{j} + 6\hat{k}\) is:
\[ \hat{a} = \frac{3}{7}\hat{i} + \frac{2}{7}\hat{j} + \frac{6}{7}\hat{k} \]