40. The line $l$ passes through the origin, its direction is determined by the unit vector $\vec{e}$, which forms angles $\alpha, \beta$, and $\gamma$ with the coordinate axes $O x, O y, O z$. Any point $M$ on this line defines the radius vector $\vec{r}=\overrightarrow{O M}$, which is collinear with the unit vector $\vec{e}$. The condition of collinearity is written as the equation of the line:

$$
\vec{r}=\overrightarrow{t e}
$$

where $t$ is a variable parameter, ranging over all real values from $-\infty$ to $+\infty$. Convert equation (1) to the coordinate form:

$$
\frac{x}{\cos \alpha}=\frac{y}{\cos \beta}=\frac{z^{1}}{\cos \gamma}
$$
40. Let $\cos \alpha, \cos \beta, \cos \gamma$ be the direction cosines of the unit vector $\vec{e}$. Since vectors $\vec{r}$ and $\vec{e}$ are collinear, from the equality $x \vec{i} +$ 132
$+y \vec{j}+z \vec{k}=t \vec{i} \cos \alpha+t \vec{j} \cos \beta+t \vec{k} \cos \gamma \quad$ we obtain: $\quad x=t \cos \alpha$, $y=t \cos \beta, z=t \cos \gamma ; \frac{x}{\cos \alpha}=\frac{y}{\cos \beta}=\frac{z}{\cos \gamma}$.