What are direction ratios

Question

By definition it is said that the numbers which are proportional to the direction cosines are called direction ratios. And what does proportional means.how it works I didn't get that

Answer 1

Let $(e_{k})_{k=1}^{n}$ denote the standard basis in $n$-dimensional Euclidean space. (For example, in three space we have $e_{1} = (1, 0, 0)$, $e_{2} = (0, 1, 0)$, and $e_{3} = (0, 0, 1)$.)

If $u = (u_{1}, u_{2}, \dots, u_{n})$ is a unit vector, its Cartesian components are its direction cosines. The name originates with the property that if $\theta_{k}$ is the small angle between $u$ and the $k$th standard basis vector, then $$ u_{k} = u \cdot e_{k} = \cos\theta_{k}. $$

A vector $x$ is proportional to $u$ if $x = tu$ for some real scalar $t$, i.e., if $x_{k} = tu_{k}$ for each $k$. In your terminology, direction ratios are the components of $x$. In case it's helpful, direction ratios are homogeneous coordinates in $(n-1)$-dimensional projective space.