The relation between direction-cosines l, m and n of a line is :

Options

A. `l^(2) + m^(2) + n^(2) = 1 `
B. `l^(2) + m^(2) + n^(2) = -1`
C. `l^(2) + m^(2) + n^(2) = 0 `
D. `l^(2) + m^(2) = n^(2)`.

Answer

To find the relation between the direction cosines \( l, m, n \) of a line, we start by understanding the definitions of direction cosines and the angles they represent.

### Step-by-Step Solution:

1. **Understanding Direction Cosines**:
- The direction cosines \( l, m, n \) of a line are defined as the cosines of the angles that the line makes with the coordinate axes.
- Let \( \alpha \) be the angle made with the x-axis, \( \beta \) with the y-axis, and \( \gamma \) with the z-axis.
- Thus, we have:
\[ l = \cos \alpha, \quad m = \cos \beta, \quad n = \cos \gamma \]

2. **Using the Property of Angles**:
- In three-dimensional geometry, the angles \( \alpha, \beta, \gamma \) are related to the direction cosines through the following equation:
\[ \cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma = 1 \]

3. **Substituting Direction Cosines**:
- By substituting the definitions of \( l, m, n \) into the equation, we get:
\[ l^2 + m^2 + n^2 = \cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma \]

4. **Final Relation**:
- Therefore, we arrive at the relation:
\[ l^2 + m^2 + n^2 = 1 \]

### Conclusion:
The relation between the direction cosines \( l, m, n \) of a line is given by:
\[ l^2 + m^2 + n^2 = 1 \]