If A=(1,2,3),B=(2,3,4) and AB is produced upto C such that 2AB=BC, then C=

Options

A. A. (5,4,6)
B. B. (4,5,6)
C. C. (2,3,4)
D. D. (1,2,3)

Answer

To solve the problem, we need to find the coordinates of point C given points A and B, and the condition that \(2AB = BC\).

### Step-by-Step Solution:

1. **Identify Points A and B**:
- Given \( A = (1, 2, 3) \)
- Given \( B = (2, 3, 4) \)

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

3. **Determine the Length of Vector AB**:
- The length of vector \( \overrightarrow{AB} \) is:
\[ |\overrightarrow{AB}| = \sqrt{(1)^2 + (1)^2 + (1)^2} = \sqrt{3} \]

4. **Calculate the Vector BC**:
- According to the problem, \( 2AB = BC \). Therefore, the vector \( \overrightarrow{BC} \) can be expressed as:
\[ \overrightarrow{BC} = 2 \cdot \overrightarrow{AB} = 2 \cdot (1, 1, 1) = (2, 2, 2) \]

5. **Find Coordinates of Point C**:
- Since \( C \) is produced from \( B \) in the direction of \( \overrightarrow{BC} \), we can find the coordinates of \( C \) as follows:
\[ C = B + \overrightarrow{BC} = (2, 3, 4) + (2, 2, 2) = (2 + 2, 3 + 2, 4 + 2) = (4, 5, 6) \]

### Final Answer:
Thus, the coordinates of point \( C \) are:
\[ C = (4, 5, 6) \]