Let `A(2,-1,4)` and `B(0,2,-3)` be the points and C be a point on AB produced such that `2AC=3AB`, then the coordinates of C are

Options

A. `(1/2,5/4,-5/4)`
B. `(-1/2,7/4,-13/4)`
C. `(6,-7,18)`
D. none of these

Explanation

To find the coordinates of point C that lies on the line segment AB produced such that \( 2AC = 3AB \), we can follow these steps:

### Step 1: Identify the Coordinates of Points A and B
The coordinates of points A and B are given as:
- \( A(2, -1, 4) \)
- \( B(0, 2, -3) \)

### Step 2: Calculate the Vector AB
The vector \( \overrightarrow{AB} \) can be calculated as:
\[ \overrightarrow{AB} = B - A = (0 - 2, 2 - (-1), -3 - 4) = (-2, 3, -7) \]

### Step 3: Express the Relationship Between AC and AB
We know from the problem statement that:
\[ 2AC = 3AB \]
This implies:
\[ AC = \frac{3}{2} AB \]

### Step 4: Find the Ratio of AC to BC
From the equation \( 2AC = 3AB \), we can derive the relationship between segments AC and BC. Rearranging gives:
\[ AC - BC = \frac{3}{2} AB \]
This means that point C divides the line segment AB externally in the ratio \( 3:1 \).

### Step 5: Use the Section Formula for External Division
When a point divides a line segment externally in the ratio \( m:n \), the coordinates can be found using the formula:
\[ C\left( \frac{m x_2 - n x_1}{m - n}, \frac{m y_2 - n y_1}{m - n}, \frac{m z_2 - n z_1}{m - n} \right) \]
Here, \( m = 3 \), \( n = 1 \), \( A(x_1, y_1, z_1) = (2, -1, 4) \), and \( B(x_2, y_2, z_2) = (0, 2, -3) \).

### Step 6: Substitute the Values into the Formula
Substituting the values into the section formula:
- For the x-coordinate:
\[ x_C = \frac{3 \cdot 0 - 1 \cdot 2}{3 - 1} = \frac{0 - 2}{2} = \frac{-2}{2} = -1 \]

- For the y-coordinate:
\[ y_C = \frac{3 \cdot 2 - 1 \cdot (-1)}{3 - 1} = \frac{6 + 1}{2} = \frac{7}{2} \]

- For the z-coordinate:
\[ z_C = \frac{3 \cdot (-3) - 1 \cdot 4}{3 - 1} = \frac{-9 - 4}{2} = \frac{-13}{2} \]

### Step 7: Write the Final Coordinates of Point C
Thus, the coordinates of point C are:
\[ C\left(-1, \frac{7}{2}, -\frac{13}{2}\right) \]

### Final Answer
The coordinates of point C are:
\[ C\left(-1, \frac{7}{2}, -\frac{13}{2}\right) \]