From the point `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 direction ratios of the line segment PO?

Options

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

Answer

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

### Step 1: Parameterize the line \( L \)
The line \( L \) can be expressed in terms of a parameter \( \lambda \):
\[ x = 2\lambda, \quad y = 3\lambda + 2, \quad z = 4\lambda + 3 \]
This gives us the coordinates of any point \( Q \) on the line in terms of \( \lambda \).

### Step 2: Find the direction ratios of the line \( L \)
From the parameterization, the direction ratios of the line \( L \) are \( (2, 3, 4) \).

### Step 3: Set up the equation for the perpendicularity condition
The direction ratios of the line segment \( PQ \) can be expressed as:
\[ PQ = (2\lambda - 3, 3\lambda + 2 + 1, 4\lambda + 3 - 11) = (2\lambda - 3, 3\lambda + 3, 4\lambda - 8) \]
For \( PQ \) to be perpendicular to \( L \), the dot product of the direction ratios must equal zero:
\[ (2, 3, 4) \cdot (2\lambda - 3, 3\lambda + 3, 4\lambda - 8) = 0 \]

### Step 4: Expand the dot product
Calculating the dot product:
\[ 2(2\lambda - 3) + 3(3\lambda + 3) + 4(4\lambda - 8) = 0 \]
Expanding this gives:
\[ 4\lambda - 6 + 9\lambda + 9 + 16\lambda - 32 = 0 \]
Combining like terms:
\[ (4\lambda + 9\lambda + 16\lambda) + (-6 + 9 - 32) = 0 \]
\[ 29\lambda - 29 = 0 \]

### Step 5: Solve for \( \lambda \)
Setting the equation to zero:
\[ 29\lambda - 29 = 0 \implies 29\lambda = 29 \implies \lambda = 1 \]

### Step 6: Find the coordinates of point \( Q \)
Substituting \( \lambda = 1 \) back into the parameterization:
\[ x = 2(1) = 2, \quad y = 3(1) + 2 = 5, \quad z = 4(1) + 3 = 7 \]
Thus, the coordinates of point \( Q \) are \( (2, 5, 7) \).

### Step 7: Calculate the direction ratios of line segment \( PQ \)
The direction ratios of line segment \( PQ \) are:
\[ PQ = (2 - 3, 5 + 1, 7 - 11) = (-1, 6, -4) \]

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

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