To solve the problem of finding the foot of the perpendicular from the point \( P(2, -1, 5) \) to the line defined by the vector equation \( \vec{r} = (11\hat{i} - 2\hat{j} - 8\hat{k}) + \lambda(10\hat{i} - 4\hat{j} - 11\hat{k}) \), we will follow these steps:
### Step 1: Identify the point on the line
The line can be expressed in parametric form as:
- \( x = 11 + 10\lambda \)
- \( y = -2 - 4\lambda \)
- \( z = -8 - 11\lambda \)
Let \( N \) be a point on the line corresponding to the parameter \( \lambda \). Thus, we can write:
\[
N(\lambda) = (11 + 10\lambda, -2 - 4\lambda, -8 - 11\lambda)
\]
### Step 2: Find the direction ratios of the line
The direction ratios of the line are given by the coefficients of \( \lambda \) in the parametric equations:
- \( d_x = 10 \)
- \( d_y = -4 \)
- \( d_z = -11 \)
### Step 3: Find the direction ratios of the perpendicular line
The direction ratios of the line \( PN \) (from point \( P \) to point \( N \)) can be expressed as:
\[
\text{Direction ratios of } PN = (11 + 10\lambda - 2, -2 - 4\lambda + 1, -8 - 11\lambda - 5)
\]
This simplifies to:
\[
(10\lambda + 9, -4\lambda - 1, -11\lambda - 13)
\]
### Step 4: Set up the scalar product equation
Since \( PN \) is perpendicular to the line, the scalar product of the direction ratios of \( PN \) and the direction ratios of the line must equal zero:
\[
(10)(10\lambda + 9) + (-4)(-4\lambda - 1) + (-11)(-11\lambda - 13) = 0
\]
### Step 5: Expand and simplify the equation
Expanding the equation:
\[
100\lambda + 90 + 16\lambda + 4 + 121\lambda + 143 = 0
\]
Combining like terms:
\[
(100 + 16 + 121)\lambda + (90 + 4 + 143) = 0
\]
\[
237\lambda + 237 = 0
\]
Thus, we find:
\[
\lambda = -1
\]
### Step 6: Substitute \( \lambda \) back to find point \( N \)
Now substitute \( \lambda = -1 \) back into the equations for \( N \):
\[
N(-1) = (11 + 10(-1), -2 - 4(-1), -8 - 11(-1))
\]
Calculating each component:
\[
N(-1) = (11 - 10, -2 + 4, -8 + 11) = (1, 2, 3)
\]
### Step 7: Find the length of the perpendicular
The length of the perpendicular \( d \) from point \( P(2, -1, 5) \) to point \( N(1, 2, 3) \) is given by the distance formula:
\[
d = \sqrt{(1 - 2)^2 + (2 - (-1))^2 + (3 - 5)^2}
\]
Calculating:
\[
d = \sqrt{(-1)^2 + (3)^2 + (-2)^2} = \sqrt{1 + 9 + 4} = \sqrt{14}
\]
### Final Answer
The foot of the perpendicular from the point \( P(2, -1, 5) \) on the line is \( N(1, 2, 3) \) and the length of the perpendicular is \( \sqrt{14} \).
