The length of the perpendicular from P(1,0,2) on the line `(x+1)/(3)=(y-2)/(-2)=(z+1)/(-1)` is

Options

A. `(3 sqrt6)/(2)`
B. `(6sqrt3)/(5)`
C. `3sqrt2`
D. `2sqrt3`

Answer

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 will follow these steps:

### Step 1: Parametrize the Line
The line can be expressed in parametric form using a parameter \( \lambda \):
- From \( \frac{x + 1}{3} = \lambda \), we get \( x = 3\lambda - 1 \).
- From \( \frac{y - 2}{-2} = \lambda \), we get \( y = -2\lambda + 2 \).
- From \( \frac{z + 1}{-1} = \lambda \), we get \( z = -\lambda - 1 \).

Thus, any point \( Q \) on the line can be represented as:
\[ Q(3\lambda - 1, -2\lambda + 2, -\lambda - 1). \]

### Step 2: Find Direction Ratios
The direction ratios of the line can be derived from the coefficients of \( \lambda \):
- Direction ratios are \( (3, -2, -1) \).

### Step 3: Find Direction Ratios of Line \( PQ \)
The direction ratios of the line segment \( PQ \) from point \( P(1, 0, 2) \) to point \( Q(3\lambda - 1, -2\lambda + 2, -\lambda - 1) \) are given by:
- \( PQ: (3\lambda - 1 - 1, -2\lambda + 2 - 0, -\lambda - 1 - 2) \)
- This simplifies to \( (3\lambda - 2, -2\lambda + 2, -\lambda - 3) \).

### Step 4: Use the Condition of Perpendicularity
For the lines \( PQ \) and the line itself to be perpendicular, the dot product of their direction ratios must be zero:
\[ (3)(3\lambda - 2) + (-2)(-2\lambda + 2) + (-1)(-\lambda - 3) = 0. \]

Expanding this gives:
\[ 9\lambda - 6 + 4\lambda - 4 + \lambda + 3 = 0. \]
Combining like terms:
\[ (9\lambda + 4\lambda + \lambda) + (-6 - 4 + 3) = 0 \implies 14\lambda - 7 = 0. \]
Thus, we find:
\[ \lambda = \frac{1}{2}. \]

### Step 5: Find the Coordinates of Point \( Q \)
Substituting \( \lambda = \frac{1}{2} \) back into the equations for \( Q \):
- \( x = 3\left(\frac{1}{2}\right) - 1 = \frac{3}{2} - 1 = \frac{1}{2} \),
- \( y = -2\left(\frac{1}{2}\right) + 2 = -1 + 2 = 1 \),
- \( z = -\left(\frac{1}{2}\right) - 1 = -\frac{1}{2} - 1 = -\frac{3}{2} \).

Thus, the coordinates of point \( Q \) are \( \left(\frac{1}{2}, 1, -\frac{3}{2}\right) \).

### Step 6: Calculate the Distance \( PQ \)
Now, we can calculate the distance \( PQ \) using the distance formula:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}, \]
where \( P(1, 0, 2) \) and \( Q\left(\frac{1}{2}, 1, -\frac{3}{2}\right) \):
\[ d = \sqrt{\left(1 - \frac{1}{2}\right)^2 + (0 - 1)^2 + \left(2 - \left(-\frac{3}{2}\right)\right)^2}. \]
Calculating each term:
- \( \left(1 - \frac{1}{2}\right)^2 = \left(\frac{1}{2}\right)^2 = \frac{1}{4} \),
- \( (0 - 1)^2 = (-1)^2 = 1 \),
- \( \left(2 + \frac{3}{2}\right)^2 = \left(\frac{4}{2} + \frac{3}{2}\right)^2 = \left(\frac{7}{2}\right)^2 = \frac{49}{4} \).

Thus,
\[ d = \sqrt{\frac{1}{4} + 1 + \frac{49}{4}} = \sqrt{\frac{1 + 4 + 49}{4}} = \sqrt{\frac{54}{4}} = \sqrt{\frac{27}{2}} = \frac{3\sqrt{6}}{2}. \]

### Final Answer
The length of the perpendicular from point \( P(1, 0, 2) \) to the line is \( \frac{3\sqrt{6}}{2} \).