Shortest distance between the lines `(x-1)/1=(y-1)/1=(z-1)/1a n d(x-2)/1=(y-3)/1=(z-4)/1` is equal to

Options

A. `sqrt14`
B. `sqrt7`
C. `sqrt2`
D. none of these

Explanation

To find the shortest distance between the two given lines, we can use the formula for the distance between two parallel lines in three-dimensional space. The lines are given in symmetric form:

1. Line 1: \((x-1)/1 = (y-1)/1 = (z-1)/1\)
2. Line 2: \((x-2)/1 = (y-3)/1 = (z-4)/1\)

### Step 1: Identify Points and Direction Ratios
From the equations of the lines, we can identify:
- For Line 1, a point on the line \(P_1(1, 1, 1)\) and the direction ratios \(a_1 = 1, b_1 = 1, c_1 = 1\).
- For Line 2, a point on the line \(P_2(2, 3, 4)\) and the direction ratios \(a_2 = 1, b_2 = 1, c_2 = 1\).

### Step 2: Check if Lines are Parallel
Since the direction ratios of both lines are the same \((1, 1, 1)\), we conclude that the lines are parallel.

### Step 3: Use the Distance Formula for Parallel Lines
The formula for the shortest distance \(d\) between two parallel lines is given by:

\[ d = \frac{|(P_2 - P_1) \cdot (a \times b)|}{|a|} \]

Where:
- \(P_1\) and \(P_2\) are points on the two lines.
- \(a\) is the direction vector of the first line.
- \(b\) is the direction vector of the second line (which is the same in this case).

### Step 4: Calculate the Vector \(P_2 - P_1\)
\[ P_2 - P_1 = (2 - 1, 3 - 1, 4 - 1) = (1, 2, 3) \]

### Step 5: Calculate the Cross Product \(a \times b\)
Since both direction vectors are the same, we can denote \(a = (1, 1, 1)\).

The cross product \(a \times a\) is zero, but since we are looking for the distance between two parallel lines, we can directly use the formula for the distance.

### Step 6: Calculate the Magnitude of the Direction Vector
\[ |a| = \sqrt{1^2 + 1^2 + 1^2} = \sqrt{3} \]

### Step 7: Calculate the Distance
Now, we can use the formula for distance:

1. Calculate the numerator:
\[ |(P_2 - P_1) \cdot a| = |(1, 2, 3) \cdot (1, 1, 1)| = |1 \cdot 1 + 2 \cdot 1 + 3 \cdot 1| = |1 + 2 + 3| = |6| = 6 \]

2. Now substitute into the distance formula:
\[ d = \frac{6}{\sqrt{3}} = 2\sqrt{3} \]

### Final Answer
The shortest distance between the two lines is \(2\sqrt{3}\).

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