Find the co-ordinates of the foot of perpendicular and length of perpendicular drawn from point `(hati+6hatj+3hatk)` to the line `vecr = hatj+2hatk+lambda(hati+2hatj+3hatk)`.

Answer

To find the coordinates of the foot of the perpendicular and the length of the perpendicular drawn from the point \( P(1, 6, 3) \) to the line given by the equation \( \vec{r} = \hat{j} + 2\hat{k} + \lambda(\hat{i} + 2\hat{j} + 3\hat{k}) \), we will follow these steps:

### Step 1: Identify the position vector of the point and the line
The position vector of point \( P \) is:
\[ \vec{P} = \hat{i} + 6\hat{j} + 3\hat{k} \]

The line can be expressed in parametric form as:
\[ \vec{r} = \hat{j} + 2\hat{k} + \lambda(\hat{i} + 2\hat{j} + 3\hat{k}) \]
This can be rewritten as:
\[ \vec{r} = \lambda \hat{i} + (1 + 2\lambda) \hat{j} + (2 + 3\lambda) \hat{k} \]

### Step 2: Determine the coordinates of the foot of the perpendicular
Let the coordinates of the foot of the perpendicular \( Q \) be \( (x_1, y_1, z_1) \). Since \( Q \) lies on the line, we have:
\[ x_1 = \lambda, \quad y_1 = 1 + 2\lambda, \quad z_1 = 2 + 3\lambda \]

### Step 3: Find the vector \( \vec{PQ} \)
The vector \( \vec{PQ} \) from point \( P \) to point \( Q \) is given by:
\[ \vec{PQ} = \vec{Q} - \vec{P} = (x_1 - 1)\hat{i} + (y_1 - 6)\hat{j} + (z_1 - 3)\hat{k} \]
Substituting the values of \( x_1, y_1, z_1 \):
\[ \vec{PQ} = (\lambda - 1)\hat{i} + ((1 + 2\lambda) - 6)\hat{j} + ((2 + 3\lambda) - 3)\hat{k} \]
This simplifies to:
\[ \vec{PQ} = (\lambda - 1)\hat{i} + (2\lambda - 5)\hat{j} + (3\lambda - 1)\hat{k} \]

### Step 4: Find the direction vector of the line
The direction vector \( \vec{b} \) of the line is:
\[ \vec{b} = \hat{i} + 2\hat{j} + 3\hat{k} \]

### Step 5: Set up the perpendicularity condition
Since \( \vec{PQ} \) is perpendicular to \( \vec{b} \), we have:
\[ \vec{PQ} \cdot \vec{b} = 0 \]
Calculating the dot product:
\[ (\lambda - 1) \cdot 1 + (2\lambda - 5) \cdot 2 + (3\lambda - 1) \cdot 3 = 0 \]
Expanding this gives:
\[ \lambda - 1 + 4\lambda - 10 + 9\lambda - 3 = 0 \]
Combining like terms:
\[ 14\lambda - 14 = 0 \]
Thus, we find:
\[ \lambda = 1 \]

### Step 6: Substitute \( \lambda \) back to find coordinates of \( Q \)
Substituting \( \lambda = 1 \) into the equations for \( Q \):
\[ x_1 = 1, \quad y_1 = 1 + 2(1) = 3, \quad z_1 = 2 + 3(1) = 5 \]
Thus, the coordinates of the foot of the perpendicular \( Q \) are:
\[ Q(1, 3, 5) \]

### Step 7: Calculate the length of the perpendicular
The length \( L \) of the perpendicular \( PQ \) is given by the distance formula:
\[ L = \sqrt{(x_1 - 1)^2 + (y_1 - 6)^2 + (z_1 - 3)^2} \]
Substituting the coordinates:
\[ L = \sqrt{(1 - 1)^2 + (3 - 6)^2 + (5 - 3)^2} = \sqrt{0 + (-3)^2 + 2^2} = \sqrt{9 + 4} = \sqrt{13} \]

### Final Result
The coordinates of the foot of the perpendicular are \( Q(1, 3, 5) \) and the length of the perpendicular is \( \sqrt{13} \).

---