Straight Lines

The equation of the line is given by the symmetric equation:

, where is a parameter.

This represents a line in three-dimensional space, with the direction ratios provided by the denominators of each fraction (i.e., coefficients of , , and ) as they relate to the parameter . These direction ratios are , , and .

The direction ratios of a line in three dimensions can be arbitrarily scaled. Therefore, any set of direction ratios is valid if it is a scalar multiple of .

Let's examine the given options:

1. : This is the original set of direction ratios.
2. : This is a scalar multiple of the original by .
3. : This cannot be a scalar multiple of the original because changing the sign of only one component breaks the proportional relationship that defines direction ratios.
4. : This is a scalar multiple of the original by .

Thus, the option cannot be the direction ratios of the line as it doesn't maintain the required proportional transformations that can exist between direction ratios. This inconsistent alteration indicates it does not align with the line's direction as derived from the given symmetric equation.