Decompose the vector `6 hati - 3 hatj - 6 hatk` into vectors which are parallel and perpendicular to the vector `hati + hatj + hatk.`

Answer

To decompose the vector \( \mathbf{A} = 6 \hat{i} - 3 \hat{j} - 6 \hat{k} \) into components that are parallel and perpendicular to the vector \( \mathbf{B} = \hat{i} + \hat{j} + \hat{k} \), we can follow these steps:

### Step 1: Identify the vectors
We have:
- \( \mathbf{A} = 6 \hat{i} - 3 \hat{j} - 6 \hat{k} \)
- \( \mathbf{B} = \hat{i} + \hat{j} + \hat{k} \)

### Step 2: Find the unit vector in the direction of \( \mathbf{B} \)
To find the unit vector \( \hat{b} \) in the direction of \( \mathbf{B} \):
\[ \hat{b} = \frac{\mathbf{B}}{|\mathbf{B}|} \]
First, calculate the magnitude of \( \mathbf{B} \):
\[ |\mathbf{B}| = \sqrt{1^2 + 1^2 + 1^2} = \sqrt{3} \]
Thus, the unit vector \( \hat{b} \) is:
\[ \hat{b} = \frac{1}{\sqrt{3}}(\hat{i} + \hat{j} + \hat{k}) \]

### Step 3: Find the projection of \( \mathbf{A} \) onto \( \hat{b} \)
The projection \( \mathbf{C} \) of \( \mathbf{A} \) onto \( \hat{b} \) is given by:
\[ \mathbf{C} = \left( \frac{\mathbf{A} \cdot \hat{b}}{|\hat{b}|^2} \right) \hat{b} \]
Calculating \( \mathbf{A} \cdot \hat{b} \):
\[ \mathbf{A} \cdot \hat{b} = (6 \hat{i} - 3 \hat{j} - 6 \hat{k}) \cdot \left( \frac{1}{\sqrt{3}}(\hat{i} + \hat{j} + \hat{k}) \right) = \frac{1}{\sqrt{3}}(6 - 3 - 6) = \frac{-3}{\sqrt{3}} = -\sqrt{3} \]
Since \( |\hat{b}|^2 = 1 \), we have:
\[ \mathbf{C} = -\sqrt{3} \cdot \hat{b} = -\sqrt{3} \cdot \frac{1}{\sqrt{3}}(\hat{i} + \hat{j} + \hat{k}) = -(\hat{i} + \hat{j} + \hat{k}) = -\hat{i} - \hat{j} - \hat{k} \]

### Step 4: Find the perpendicular component \( \mathbf{D} \)
The vector \( \mathbf{D} \) that is perpendicular to \( \hat{b} \) can be found using:
\[ \mathbf{D} = \mathbf{A} - \mathbf{C} \]
Substituting the values:
\[ \mathbf{D} = (6 \hat{i} - 3 \hat{j} - 6 \hat{k}) - (-\hat{i} - \hat{j} - \hat{k}) = (6 + 1) \hat{i} + (-3 + 1) \hat{j} + (-6 + 1) \hat{k} = 7 \hat{i} - 2 \hat{j} - 5 \hat{k} \]

### Final Result
Thus, the decomposition of the vector \( \mathbf{A} \) into components parallel and perpendicular to \( \mathbf{B} \) is:
- Parallel component \( \mathbf{C} = -\hat{i} - \hat{j} - \hat{k} \)
- Perpendicular component \( \mathbf{D} = 7 \hat{i} - 2 \hat{j} - 5 \hat{k} \)