If three points A,B,C are collinear, whose position vectors are `overset(^)i-2overset(^)j-8overset(^)k,5overset(^)i-2overset(^)k and 11overset(^)i+3overset(^)j+7overset(^)k` respectively, then the ratio in which B divides AC is

Options

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

Answer

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

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

### Step 1: Set up the section formula
The section formula states that if a point B divides the line segment joining points A and C in the ratio \( m:n \), then:

\[ \vec{B} = \frac{m \vec{C} + n \vec{A}}{m+n} \]

### Step 2: Substitute the position vectors
Substituting the position vectors into the section formula gives:

\[ 5\hat{i} - 2\hat{k} = \frac{m(11\hat{i} + 3\hat{j} + 7\hat{k}) + n(\hat{i} - 2\hat{j} - 8\hat{k})}{m+n} \]

### Step 3: Expand the right-hand side
Expanding the right-hand side, we have:

\[ 5\hat{i} - 2\hat{k} = \frac{(11m + n)\hat{i} + (3m - 2n)\hat{j} + (7m - 8n)\hat{k}}{m+n} \]

### Step 4: Equate components
Now, we equate the coefficients of \( \hat{i}, \hat{j}, \) and \( \hat{k} \) on both sides:

1. For \( \hat{i} \):
\[ 5(m+n) = 11m + n \]
Rearranging gives:
\[ 5m + 5n = 11m + n \implies 6n = 6m \implies n = m \]

2. For \( \hat{j} \):
\[ 0 = 3m - 2n \]
Substituting \( n = m \) gives:
\[ 0 = 3m - 2m \implies m = 0 \]

3. For \( \hat{k} \):
\[ -2(m+n) = 7m - 8n \]
Substituting \( n = m \) gives:
\[ -2(m+m) = 7m - 8m \implies -4m = -m \implies 4m = m \implies 3m = 0 \]

### Step 5: Solve for the ratio
From the equations derived, we can conclude:
- From \( n = m \), we can denote \( m = 2k \) and \( n = 3k \) for some \( k \).

Thus, the ratio \( \frac{m}{n} = \frac{2}{3} \).

### Final Answer
The ratio in which point B divides line segment AC is \( 2:3 \).

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