Find the co-ordiantes of the foot of perpendicular drawn from the point `(1,2,3)` to the line `(x-6)/(3) = (y-7)/(2) = (z-7)/(-2)`.

Answer

To find the coordinates of the foot of the perpendicular drawn from the point \( P(1, 2, 3) \) to the line given by the equations \( \frac{x-6}{3} = \frac{y-7}{2} = \frac{z-7}{-2} \), we can follow these steps:

### Step 1: Parametrize the Line
The line can be expressed in parametric form. Let \( t \) be the parameter:
- \( x = 3t + 6 \)
- \( y = 2t + 7 \)
- \( z = -2t + 7 \)

### Step 2: Define the Foot of the Perpendicular
Let the coordinates of the foot of the perpendicular be \( Q(x, y, z) \). Since \( Q \) lies on the line, we can express its coordinates in terms of \( t \):
- \( Q(t) = (3t + 6, 2t + 7, -2t + 7) \)

### Step 3: Find the Direction Ratios
The direction ratios of the line are given by the coefficients of \( t \) in the parametric equations:
- Direction ratios of the line \( L \): \( (3, 2, -2) \)

The direction ratios of the line segment \( PQ \) (from point \( P \) to point \( Q \)) can be calculated as:
- \( PQ = Q - P = (3t + 6 - 1, 2t + 7 - 2, -2t + 7 - 3) = (3t + 5, 2t + 5, -2t + 4) \)

### Step 4: Use the Perpendicularity Condition
For the lines \( PQ \) and \( L \) to be perpendicular, the dot product of their direction ratios must equal zero:
\[ (3, 2, -2) \cdot (3t + 5, 2t + 5, -2t + 4) = 0 \]
Calculating the dot product:
\[ 3(3t + 5) + 2(2t + 5) - 2(-2t + 4) = 0 \]
Expanding this gives:
\[ 9t + 15 + 4t + 10 + 4t - 8 = 0 \]
Combining like terms:
\[ (9t + 4t + 4t) + (15 + 10 - 8) = 0 \implies 17t + 17 = 0 \]
Thus, we find:
\[ t = -1 \]

### Step 5: Substitute \( t \) Back to Find \( Q \)
Now substitute \( t = -1 \) back into the parametric equations to find the coordinates of \( Q \):
- \( x = 3(-1) + 6 = 3 \)
- \( y = 2(-1) + 7 = 5 \)
- \( z = -2(-1) + 7 = 9 \)

So, the coordinates of the foot of the perpendicular \( Q \) are \( (3, 5, 9) \).

### Final Answer
The coordinates of the foot of the perpendicular from the point \( (1, 2, 3) \) to the line are:
\[ \boxed{(3, 5, 9)} \]