algebraic representation of a line in 3d Is an algebraic representation of a line in 3d possible, or there can be only a parametric one? 

Suppose we have described our line $\vec{r}$ in 3D using parametric form 
$$\vec{r}(t) = r_0 + t\vec{v}$$
where $t$ is our parameter.
It follows that $\vec{r}(t) = \langle x(t),y(t),z(t) \rangle$ is the position at time $t$, $r_0 = \langle x_0,y_0,z_0 \rangle$ is some point on the line, and  $\vec{v}=\langle a,b,c \rangle$ is a vector in the direction of the line.
Equating the left and right-hand side components of the equation, we find that
$$x(t) = x_0 + at$$
$$y(t) = y_0 + bt$$
$$z(t) = z_0 + ct$$
We can then eliminate the parameter $t$, which gives the following Cartesian representation of the line
$$ t = \frac{x-x_0}{a} =  \frac{y-y_0}{b}= \frac{z-z_0}{c}$$
