Find the foot of the perpendicular drawn from the point `2 hat i- hat j+5 hat k` to the line ` vec r=(11 hat i-2 hat j-8 hat k)+lambda(10 hat i-4 hat j-11 hat k)dot` Also find the length of the perpendicular.

Explanation

To find the foot of the perpendicular drawn from the point \( \vec{P} = 2\hat{i} - \hat{j} + 5\hat{k} \) to the line given 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 can follow these steps:

### Step 1: Identify the components of the line and the point
The line can be expressed in terms of a point on the line \( \vec{A} = 11\hat{i} - 2\hat{j} - 8\hat{k} \) and a direction vector \( \vec{B} = 10\hat{i} - 4\hat{j} - 11\hat{k} \). The point \( \vec{P} \) is given as \( 2\hat{i} - \hat{j} + 5\hat{k} \).

### Step 2: Write the position vector of the foot of the perpendicular
Let \( \vec{L} \) be the foot of the perpendicular from point \( \vec{P} \) to the line. The position vector of \( \vec{L} \) can be expressed as:
\[
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