If three points A,B and C are collinear, whose position vectors are `hati-2hatj-8hatk,5hati-2hatk and 11hati+3hatj+7hatk` respectively, then the ratio in which B divides AC is

Options

A. `1:2`
B. `2:3`
C. `2:1`
D. `1:1`

Explanation

To find the ratio in which point B divides the line segment AC, we will use the section formula. Let's denote the position vectors of points A, B, and C as follows:

- Position vector of A: **A** = \( \hat{i} - 2\hat{j} - 8\hat{k} \)
- Position vector of B: **B** = \( 5\hat{i} - 2\hat{k} \)
- Position vector of C: **C** = \( 11\hat{i} + 3\hat{j} + 7\hat{k} \)

### Step 1: Express the position vectors in terms of coordinates
We can express the position vectors in coordinate form:
- A = (1, -2, -8)
- B = (5, 0, -2)
- C = (11, 3, 7)

### Step 2: Use the section formula
Let B divide AC in the ratio \( \lambda : 1 \). According to the section formula, the position vector of point B can be expressed as:
\[ \vec{B} = \frac{\lambda \vec{C} + 1 \vec{A}}{\lambda + 1} \]

### Step 3: Substitute the position vectors
Substituting the position vectors of A and C into the equation:
\[ 5\hat{i} - 2\hat{k} = \frac{\lambda (11\hat{i} + 3\hat{j} + 7\hat{k}) + 1(\hat{i} - 2\hat{j} - 8\hat{k})}{\lambda + 1} \]

### Step 4: Separate the components
This gives us three equations, one for each component (i, j, k):
1. For the i-component:
\[ 5 = \frac{11\lambda + 1}{\lambda + 1} \]
2. For the j-component:
\[ 0 = \frac{3\lambda - 2}{\lambda + 1} \]
3. For the k-component:
\[ -2 = \frac{7\lambda - 8}{\lambda + 1} \]

### Step 5: Solve the j-component equation
From the j-component equation:
\[ 0 = 3\lambda - 2 \implies 3\lambda = 2 \implies \lambda = \frac{2}{3} \]

### Step 6: Find the ratio
The ratio in which B divides AC is \( \lambda : 1 = \frac{2}{3} : 1 \). To express this in a more standard form, we can multiply both sides by 3:
\[ 2 : 3 \]

### Conclusion
Thus, the ratio in which point B divides the line segment AC is \( 2 : 3 \).

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