To find the length of the perpendicular from the point \( P(1, 0, 2) \) to the line given by the equations \( \frac{x+1}{3} = \frac{y-2}{-2} = \frac{z+1}{-1} \), we can follow these steps:
### Step 1: Parametrize the Line
The line can be expressed in parametric form. Let \( t \) be the parameter such that:
\[
x = 3t - 1, \quad y = -2t + 2, \quad z = -t - 1
\]
This gives us a point on the line as \( M(3t - 1, -2t + 2, -t - 1) \).
### Step 2: Find the Direction Ratios of the Line
The direction ratios of the line can be derived from the coefficients of \( t \) in the parametric equations:
\[
\text{Direction Ratios} = (3, -2, -1)
\]
### Step 3: Find the Direction Ratios of PM
The vector \( PM \) from point \( P(1, 0, 2) \) to point \( M(3t - 1, -2t + 2, -t - 1) \) is given by:
\[
PM = (3t - 1 - 1, -2t + 2 - 0, -t - 1 - 2) = (3t - 2, -2t + 2, -t - 3)
\]
### Step 4: Set Up the Perpendicular Condition
Since \( PM \) is perpendicular to the direction ratios of the line, we can use the dot product:
\[
(3, -2, -1) \cdot (3t - 2, -2t + 2, -t - 3) = 0
\]
Calculating the dot product:
\[
3(3t - 2) + (-2)(-2t + 2) + (-1)(-t - 3) = 0
\]
Expanding this gives:
\[
9t - 6 + 4t - 4 + t + 3 = 0
\]
Combining like terms:
\[
14t - 7 = 0
\]
Solving for \( t \):
\[
t = \frac{1}{2}
\]
### Step 5: Find the Coordinates of Point M
Substituting \( t = \frac{1}{2} \) back into the parametric equations:
\[
x = 3\left(\frac{1}{2}\right) - 1 = \frac{3}{2} - 1 = \frac{1}{2}
\]
\[
y = -2\left(\frac{1}{2}\right) + 2 = -1 + 2 = 1
\]
\[
z = -\left(\frac{1}{2}\right) - 1 = -\frac{1}{2} - 1 = -\frac{3}{2}
\]
Thus, the coordinates of point \( M \) are \( \left(\frac{1}{2}, 1, -\frac{3}{2}\right) \).
### Step 6: Calculate the Length of PM
Now we can calculate the length of \( PM \):
\[
PM = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}
\]
Substituting \( P(1, 0, 2) \) and \( M\left(\frac{1}{2}, 1, -\frac{3}{2}\right) \):
\[
PM = \sqrt{\left(\frac{1}{2} - 1\right)^2 + (1 - 0)^2 + \left(-\frac{3}{2} - 2\right)^2}
\]
Calculating each term:
\[
= \sqrt{\left(-\frac{1}{2}\right)^2 + 1^2 + \left(-\frac{7}{2}\right)^2}
\]
\[
= \sqrt{\frac{1}{4} + 1 + \frac{49}{4}} = \sqrt{\frac{1 + 4 + 49}{4}} = \sqrt{\frac{54}{4}} = \sqrt{\frac{27}{2}} = \frac{3\sqrt{6}}{2}
\]
### Final Answer
Thus, the length of the perpendicular from point \( P \) to the line is:
\[
\frac{3\sqrt{6}}{2}
\]
