Distance of the point `P(x_2, y_2, z_2)` from the line `(x-x_1)/l=(y-y_1)/m=(z-z_1)/n`, where `l,m,n` are the direction cosines of the line, is

Options

A. `sqrt(l^(2)(x_(2)-x_(1))^(2)+m^(2)(y_(2)-y_(1))^(2)+n^(2)(z_(2)-z_(1)^(2))`
B. `|l(x_(2)-x_(1))+m(y_(2)-y_(1))+n(z_(2)-z_(1))|`
C. `sqrt((x_(2)-x_(1))^(2)+(y_(2)-y_(1))^(2)+(z_(2)-z_(1))^(2)-{l(x_(2)-x_(1))+m(y_(2)-y_(1)+n(z_(2)-z_(1))}^(2)`
D. none of these

Explanation

To find the distance of the point \( P(x_2, y_2, z_2) \) from the line given by the equation

\[ \frac{x - x_1}{l} = \frac{y - y_1}{m} = \frac{z - z_1}{n} \]

where \( l, m, n \) are the direction cosines of the line, we can follow these steps:

### Step 1: Understand the Geometry
The distance from a point to a line in space is the length of the perpendicular dropped from the point to the line. We can visualize the line and the point in three-dimensional space.

### Step 2: Define the Position Vectors
Let the position vector of point \( P \) be:

\[ \vec{P} = (x_2, y_2, z_2) \]

Let the position vector of a point \( M_1 \) on the line be:

\[ \vec{M_1} = (x_1, y_1, z_1) \]

The direction vector of the line, based on the direction cosines, can be represented as:

\[ \vec{s} = (l, m, n) \]

### Step 3: Form the Vector from \( M_1 \) to \( P \)
The vector from point \( M_1 \) to point \( P \) is given by:

\[ \vec{PM_1} = \vec{P} - \vec{M_1} = (x_2 - x_1, y_2 - y_1, z_2 - z_1) \]

### Step 4: Calculate the Cross Product
To find the distance, we need to calculate the cross product of the vector \( \vec{PM_1} \) and the direction vector \( \vec{s} \):

\[ \vec{PM_1} \times \vec{s} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ x_2 - x_1 & y_2 - y_1 & z_2 - z_1 \\ l & m & n \end{vmatrix} \]

### Step 5: Find the Magnitude of the Cross Product
The magnitude of the cross product gives the area of the parallelogram formed by the two vectors. The magnitude is calculated as:

\[ |\vec{PM_1} \times \vec{s}| = \sqrt{(y_2 - y_1)n - (z_2 - z_1)m}^2 + (z_2 - z_1)l - (x_2 - x_1)n^2 + (x_2 - x_1)m - (y_2 - y_1)l^2 \]

### Step 6: Calculate the Distance
The distance \( d \) from the point \( P \) to the line is given by the formula:

\[ d = \frac{|\vec{PM_1} \times \vec{s}|}{|\vec{s}|} \]

Where \( |\vec{s}| = \sqrt{l^2 + m^2 + n^2} \).

### Final Expression
Thus, the distance \( d \) can be expressed as:

\[ d = \frac{|\vec{PM_1} \times \vec{s}|}{\sqrt{l^2 + m^2 + n^2}} \]