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.

Answer

To solve the problem of finding the foot of the perpendicular from the point \( P(2\hat{i} - \hat{j} + 5\hat{k}) \) 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: Understand the line equation
The line can be expressed in terms of a point on the line and a direction vector. The point on the line is \( A(11, -2, -8) \) and the direction vector is \( \vec{b} = (10\hat{i} - 4\hat{j} - 11\hat{k}) \).

### Step 2: Write the position vector of the foot of the perpendicular
Let \( M \) be the foot of the perpendicular from point \( P \) to the line. The position vector of point \( M \) can be expressed as:
\[ \vec{m} = (11 + 10\lambda)\hat{i} + (-2 - 4\lambda)\hat{j} + (-8 - 11\lambda)\hat{k} \]

### Step 3: Find the vector \( \vec{PM} \)
The vector \( \vec{PM} \) from point \( P \) to point \( M \) is given by:
\[ \vec{PM} = \vec{m} - \vec{p} = \left((11 + 10\lambda - 2)\hat{i} + (-2 - 4\lambda + 1)\hat{j} + (-8 - 11\lambda - 5)\hat{k}\right) \]
This simplifies to:
\[ \vec{PM} = (9 + 10\lambda)\hat{i} + (-1 - 4\lambda)\hat{j} + (-13 - 11\lambda)\hat{k} \]

### Step 4: Set up the perpendicularity condition
Since \( \vec{PM} \) is perpendicular to the direction vector \( \vec{b} \), we have:
\[ \vec{PM} \cdot \vec{b} = 0 \]
Calculating the dot product:
\[ (9 + 10\lambda) \cdot 10 + (-1 - 4\lambda) \cdot (-4) + (-13 - 11\lambda) \cdot (-11) = 0 \]
Expanding this gives:
\[ 90 + 100\lambda + 4 + 16\lambda + 143 + 121\lambda = 0 \]
Combining like terms:
\[ (100 + 16 + 121)\lambda + (90 + 4 + 143) = 0 \]
This simplifies to:
\[ 237\lambda + 237 = 0 \]
Thus, we find:
\[ \lambda = -1 \]

### Step 5: Find the coordinates of the foot of the perpendicular
Substituting \( \lambda = -1 \) back into the equation for \( \vec{m} \):
\[ \vec{m} = (11 + 10(-1))\hat{i} + (-2 - 4(-1))\hat{j} + (-8 - 11(-1))\hat{k} \]
This simplifies to:
\[ \vec{m} = (11 - 10)\hat{i} + (-2 + 4)\hat{j} + (-8 + 11)\hat{k} = 1\hat{i} + 2\hat{j} + 3\hat{k} \]
Thus, the coordinates of the foot of the perpendicular \( M \) are \( (1, 2, 3) \).

### Step 6: Calculate the length of the perpendicular
The length of the perpendicular \( PM \) is given by:
\[ \text{Length} = |\vec{PM}| = \sqrt{(1 - 2)^2 + (2 + 1)^2 + (3 - 5)^2} \]
Calculating this gives:
\[ = \sqrt{(-1)^2 + (3)^2 + (-2)^2} = \sqrt{1 + 9 + 4} = \sqrt{14} \]

### Final Answer
The foot of the perpendicular is at the point \( (1, 2, 3) \) and the length of the perpendicular is \( \sqrt{14} \).