1. **Stating the problem:** We want to understand what direction ratios are in the context of vectors and lines in geometry. 2. **Definition:** Direction ratios are a set of three numbers that are proportional to the direction cosines of a line in 3D space. They represent the direction of the line but are not necessarily unit vectors. 3. **Formula and explanation:** If a line has direction cosines $l, m, n$, then its direction ratios can be any set of numbers $a, b, c$ such that $$\frac{a}{l} = \frac{b}{m} = \frac{c}{n}.$$ These ratios give the direction of the line. 4. **Important rules:** - Direction ratios are not unique; multiplying all three by the same nonzero scalar gives another valid set. - They are used to describe the orientation of a line without specifying length. 5. **Example:** If a line has direction cosines $\left(\frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}\right)$, then direction ratios could be $(1,1,1)$ because $$\frac{1}{\frac{1}{\sqrt{3}}} = \sqrt{3}$$ is the same for all. 6. **Summary:** Direction ratios are proportional values that indicate the direction of a line in space, useful for vector and line equations.