If `vecA= 6hati- 6hatj+5hatk and vecB= hati+ 2hatj-hatk`, then find a unit vector parallel to the resultant of `veca & vecB`.

Explanation

To find a unit vector parallel to the resultant of the vectors \(\vec{A}\) and \(\vec{B}\), we will follow these steps:

### Step 1: Write down the vectors
Given:
\[ \vec{A} = 6\hat{i} - 6\hat{j} + 5\hat{k} \]
\[ \vec{B} = \hat{i} + 2\hat{j} - \hat{k} \]

### Step 2: Find the resultant vector \(\vec{R}\)
The resultant vector \(\vec{R}\) is the sum of \(\vec{A}\) and \(\vec{B}\):
\[ \vec{R} = \vec{A} + \vec{B} \]
Calculating the components:
\[ \vec{R} = (6\hat{i} + \hat{i}) + (-6\hat{j} + 2\hat{j}) + (5\hat{k} - \hat{k}) \]
\[ \vec{R} = (6 + 1)\hat{i} + (-6 + 2)\hat{j} + (5 - 1)\hat{k} \]
\[ \vec{R} = 7\hat{i} - 4\hat{j} + 4\hat{k} \]

### Step 3: Calculate the magnitude of the resultant vector \(|\vec{R}|\)
The magnitude of \(\vec{R}\) is given by:
\[ |\vec{R}| = \sqrt{(7)^2 + (-4)^2 + (4)^2} \]
Calculating each term:
\[ |\vec{R}| = \sqrt{49 + 16 + 16} \]
\[ |\vec{R}| = \sqrt{49 + 32} = \sqrt{81} = 9 \]

### Step 4: Find the unit vector \(\hat{r}\) parallel to \(\vec{R}\)
The unit vector \(\hat{r}\) is given by:
\[ \hat{r} = \frac{\vec{R}}{|\vec{R}|} \]
Substituting \(\vec{R}\) and its magnitude:
\[ \hat{r} = \frac{7\hat{i} - 4\hat{j} + 4\hat{k}}{9} \]
This simplifies to:
\[ \hat{r} = \frac{7}{9}\hat{i} - \frac{4}{9}\hat{j} + \frac{4}{9}\hat{k} \]

### Final Answer
The unit vector parallel to the resultant of \(\vec{A}\) and \(\vec{B}\) is:
\[ \hat{r} = \frac{7}{9}\hat{i} - \frac{4}{9}\hat{j} + \frac{4}{9}\hat{k} \]