The points A,B and C with position vectors `3 hati - y hatj + 2hatk, 5 hati - hatj + hatk and 3x hati+ 3 hatj- hatk` are collinear. Find the values of x and y and also the ratio in which the point B divides AC.

Answer

To solve the problem step by step, we will find the values of \( x \) and \( y \) such that the points \( A \), \( B \), and \( C \) are collinear, and then we will determine the ratio in which point \( B \) divides the line segment \( AC \).

### Step 1: Identify the position vectors
The position vectors of points \( A \), \( B \), and \( C \) are given as:
- \( \vec{A} = 3\hat{i} - y\hat{j} + 2\hat{k} \)
- \( \vec{B} = 5\hat{i} - \hat{j} + \hat{k} \)
- \( \vec{C} = 3x\hat{i} + 3\hat{j} - \hat{k} \)

### Step 2: Find vectors \( \vec{AB} \) and \( \vec{BC} \)
The vector \( \vec{AB} \) is calculated as:
\[ \vec{AB} = \vec{B} - \vec{A} = (5\hat{i} - \hat{j} + \hat{k}) - (3\hat{i} - y\hat{j} + 2\hat{k}) \]
\[ = (5 - 3)\hat{i} + (-1 + y)\hat{j} + (1 - 2)\hat{k} = 2\hat{i} + (y - 1)\hat{j} - \hat{k} \]

The vector \( \vec{BC} \) is calculated as:
\[ \vec{BC} = \vec{C} - \vec{B} = (3x\hat{i} + 3\hat{j} - \hat{k}) - (5\hat{i} - \hat{j} + \hat{k}) \]
\[ = (3x - 5)\hat{i} + (3 + 1)\hat{j} + (-1 - 1)\hat{k} = (3x - 5)\hat{i} + 4\hat{j} - 2\hat{k} \]

### Step 3: Set up the collinearity condition
For points \( A \), \( B \), and \( C \) to be collinear, we have:
\[ \vec{AB} = \lambda \vec{BC} \]
This gives us the equations:
\[ 2\hat{i} + (y - 1)\hat{j} - \hat{k} = \lambda \left( (3x - 5)\hat{i} + 4\hat{j} - 2\hat{k} \right) \]

### Step 4: Equate the coefficients
From the above equation, we equate the coefficients of \( \hat{i} \), \( \hat{j} \), and \( \hat{k} \):
1. For \( \hat{i} \):
\[ 2 = \lambda (3x - 5) \quad \text{(1)} \]
2. For \( \hat{j} \):
\[ y - 1 = 4\lambda \quad \text{(2)} \]
3. For \( \hat{k} \):
\[ -1 = -2\lambda \quad \text{(3)} \]

### Step 5: Solve for \( \lambda \)
From equation (3):
\[ -1 = -2\lambda \implies \lambda = \frac{1}{2} \]

### Step 6: Substitute \( \lambda \) into equations (1) and (2)
Substituting \( \lambda = \frac{1}{2} \) into equation (1):
\[ 2 = \frac{1}{2}(3x - 5) \implies 4 = 3x - 5 \implies 3x = 9 \implies x = 3 \]

Substituting \( \lambda = \frac{1}{2} \) into equation (2):
\[ y - 1 = 4 \cdot \frac{1}{2} \implies y - 1 = 2 \implies y = 3 \]

### Step 7: Values of \( x \) and \( y \)
Thus, we have:
\[ x = 3, \quad y = 3 \]

### Step 8: Find the ratio in which \( B \) divides \( AC \)
Let \( B \) divide \( AC \) in the ratio \( m:n \). According to the section formula:
\[ \vec{B} = \frac{m\vec{C} + n\vec{A}}{m+n} \]
Substituting the values of \( \vec{A} \), \( \vec{B} \), and \( \vec{C} \):
\[ 5\hat{i} - \hat{j} + \hat{k} = \frac{m(9\hat{i} + 3\hat{j} - \hat{k}) + n(3\hat{i} - 3\hat{j} + 2\hat{k})}{m+n} \]

### Step 9: Equate coefficients for the ratio
Equating coefficients for \( \hat{i} \):
\[ 5(m+n) = 9m + 3n \implies 5m + 5n = 9m + 3n \implies 4m = 2n \implies \frac{m}{n} = \frac{1}{2} \]

Thus, the ratio \( m:n = 1:2 \).

### Final Answer
The values of \( x \) and \( y \) are:
\[ x = 3, \quad y = 3 \]
The ratio in which point \( B \) divides \( AC \) is:
\[ 1:2 \]