The ratio in which `hati + 2 hatj + 3 hatk ` divides the join of `-2hati + 3 hatj + 5 hatk and 7 hati - hatk ,` is

Options

A. `1:2`
B. `2:3`
C. `3:4`
D. `1:4`

Answer

To solve the problem, we need to find the ratio in which the vector \( \hat{i} + 2\hat{j} + 3\hat{k} \) divides the line segment joining the points represented by the vectors \( -2\hat{i} + 3\hat{j} + 5\hat{k} \) and \( 7\hat{i} - \hat{k} \).

### Step 1: Identify the points
Let:
- Point A = \( -2\hat{i} + 3\hat{j} + 5\hat{k} \)
- Point B = \( 7\hat{i} - \hat{k} \)
- Point P = \( \hat{i} + 2\hat{j} + 3\hat{k} \)

### Step 2: Express the position vector of point P in terms of A and B
The position vector of point P divides the line segment AB in the ratio \( k:1 \). We can express the position vector of P as:
\[ P = \frac{kB + A}{k + 1} \]

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

### Step 4: Expand and simplify
Expanding the right-hand side:
\[ \hat{i} + 2\hat{j} + 3\hat{k} = \frac{(7k - 2)\hat{i} + (3)\hat{j} + (5 - k)\hat{k}}{k + 1} \]

### Step 5: Equate coefficients
Now, we equate the coefficients of \( \hat{i} \), \( \hat{j} \), and \( \hat{k} \) from both sides.

1. For \( \hat{i} \):
\[ 1 = \frac{7k - 2}{k + 1} \implies k + 1 = 7k - 2 \implies 3 = 6k \implies k = \frac{1}{2} \]

2. For \( \hat{j} \):
\[ 2 = \frac{3}{k + 1} \implies 2(k + 1) = 3 \implies 2k + 2 = 3 \implies 2k = 1 \implies k = \frac{1}{2} \]

3. For \( \hat{k} \):
\[ 3 = \frac{5 - k}{k + 1} \implies 3(k + 1) = 5 - k \implies 3k + 3 = 5 - k \implies 4k = 2 \implies k = \frac{1}{2} \]

### Step 6: Determine the ratio
Since \( k = \frac{1}{2} \), the ratio in which point P divides the line segment AB is:
\[ \text{Ratio} = k:1 = \frac{1}{2}:1 = 1:2 \]

### Final Answer
The ratio in which \( \hat{i} + 2\hat{j} + 3\hat{k} \) divides the join of \( -2\hat{i} + 3\hat{j} + 5\hat{k} \) and \( 7\hat{i} - \hat{k} \) is \( 1:2 \).

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