why can't we eliminate the parameter of straight lines in higher dimensions Why we can't remove the parameter and find the Cartesian equation of straight lines in higher dimensions ? 
 

If we only have the two equations $$(1)\ x_1=p_1+ta_1$$
                                   $$(2)\ x_2=p_2+ta_2$$
we can eliminate $t$ and get the equation $$(x_1-p_1)a_2-(x_2-p_2)a_1=0$$
 which we can transform into $$ax_1+bx_2=c$$
 This equation is true, no matter which value $t$ has.
If we add the equation $$(3)\ x_3=p_3+ta_3$$
the value of $x_3$ depends on the value $t$, so an equation of the form
 $$ax_1+bx_2+cx_3=d$$
cannot hold for arbitary $t$ because if $x_1$ is given, $x_2$ is fixed.
Alternatively, you can think about the meaning of the normal vector. It is a vector
 which is orthognal to the line. The direction of this vector must be unique, which is 
 the case in $2$ dimensions, but not in $3$ or more. There are infinitely many vectors
 with different directions which are orthogonal to a given line in $3$ or more
 dimensions.
