The projections of a line segment on the coordinate axes are 12,4,3 respectively. The length and direction cosines of the line segment are

Options

A. `13,12/13,4/13,3/13`
B. `19,12/19,4/19,3/19`
C. `11,12/11,14/11,3/11`
D. none of these

Explanation

To solve the problem, we need to find the length and direction cosines of a line segment given its projections on the coordinate axes. The projections on the x, y, and z axes are given as 12, 4, and 3 respectively.

### Step 1: Calculate the Length of the Line Segment
The length \( L \) of the line segment can be calculated using the formula:
\[ L = \sqrt{x^2 + y^2 + z^2} \]
where \( x \), \( y \), and \( z \) are the projections on the x, y, and z axes respectively.

Substituting the given values:
\[ L = \sqrt{12^2 + 4^2 + 3^2} \]
Calculating each term:
\[ = \sqrt{144 + 16 + 9} \]
\[ = \sqrt{169} \]
\[ = 13 \]

### Step 2: Calculate the Direction Cosines
The direction cosines \( l \), \( m \), and \( n \) are given by the formulas:
\[ l = \frac{x}{L}, \quad m = \frac{y}{L}, \quad n = \frac{z}{L} \]

Substituting the values:
1. For \( l \):
\[ l = \frac{12}{13} \]

2. For \( m \):
\[ m = \frac{4}{13} \]

3. For \( n \):
\[ n = \frac{3}{13} \]

### Final Result
The length of the line segment is \( 13 \) and the direction cosines are:
- \( l = \frac{12}{13} \)
- \( m = \frac{4}{13} \)
- \( n = \frac{3}{13} \)

### Summary
- Length of the line segment: \( 13 \)
- Direction cosines: \( \left( \frac{12}{13}, \frac{4}{13}, \frac{3}{13} \right) \)