From the points `P(3, -1,11)`, a perpendicular is drawn on the line L given by the equation `x/2=(y-2)/3=(z-3)/4`. Let Q be the foot of the perpendicular.
What are the drection ratios of the line segment PQ?

Options

A. `(: 1,6,4 :)`
B. `(: -1,6,-4 :)`
C. `(: -1,-6,4 :)`
D. `(: 2,-6,4 :)`

Explanation

To find the direction ratios of the line segment PQ, where P is the point (3, -1, 11) and Q is the foot of the perpendicular from P to the line L given by the equations \( \frac{x}{2} = \frac{y-2}{3} = \frac{z-3}{4} \), we can follow these steps:

### Step 1: Identify the direction ratios of the line L
The line L can be expressed in parametric form. From the equation \( \frac{x}{2} = \frac{y-2}{3} = \frac{z-3}{4} \), we can set a parameter \( t \):
- \( x = 2t \)
- \( y = 3t + 2 \)
- \( z = 4t + 3 \)

The direction ratios of the line L are (2, 3, 4).

### Step 2: Find the coordinates of point Q
Let Q be the point on the line L corresponding to the parameter \( t \). Thus, the coordinates of Q are:
- \( Q(2t, 3t + 2, 4t + 3) \)

### Step 3: Find the vector PQ
The vector PQ can be expressed as:
\[ PQ = Q - P = (2t - 3, (3t + 2) - (-1), (4t + 3) - 11) \]
This simplifies to:
\[ PQ = (2t - 3, 3t + 3, 4t - 8) \]

### Step 4: Find the condition for PQ to be perpendicular to L
For PQ to be perpendicular to the line L, the dot product of PQ and the direction ratios of the line L must equal zero:
\[ PQ \cdot (2, 3, 4) = 0 \]
Calculating the dot product:
\[ (2t - 3) \cdot 2 + (3t + 3) \cdot 3 + (4t - 8) \cdot 4 = 0 \]
Expanding this gives:
\[ 4t - 6 + 9t + 9 + 16t - 32 = 0 \]
Combining like terms:
\[ 29t - 29 = 0 \]
Solving for \( t \):
\[ t = 1 \]

### Step 5: Substitute \( t \) back to find Q
Now substitute \( t = 1 \) back into the parametric equations for Q:
- \( Q(2 \cdot 1, 3 \cdot 1 + 2, 4 \cdot 1 + 3) = Q(2, 5, 7) \)

### Step 6: Calculate the direction ratios of PQ
Now we can find the direction ratios of the line segment PQ:
\[ PQ = Q - P = (2 - 3, 5 - (-1), 7 - 11) = (-1, 6, -4) \]

### Conclusion
The direction ratios of the line segment PQ are \((-1, 6, -4)\).