Vector equation and parametric equation for a line segment Suppose that $P = (1,1,7)$ and $Q = (8,6,1)$. Inside parenthesis are x-coordinate value, y-coordinate and z-coordinate. The question is to find vector and parametric equation for a line segment. 
Now, I used equation that goes like $(1-t)<1,1,7>+t<8,6,1>, where, 0\leq t\leq 1$. 
But, the answer says that I am wrong. So, I am wondering what is going on. 
Also, why do we write parametric equations? Are they really different from vector functions anyway? 
If vector function is $<2t,3t,1>$, according to my knowledge, parametric equations would be $x(t)=2t, y(t)=3t, z(t)=1$ ....

Your answer is correct. You can confirm this just by checking the values you get at $t=0$ and $t=1$. If you get $P$ and $Q$ respectively, then your answer works.
But there are an infinite number of parametric equations that describe a line through $P$ and $Q$. Maybe this explains the discrepancy.
You are correct that the vector-valued function:
$$
\mathbf{x}(t) = (2t, 3t, 1)
$$
is essentially the same thing as the function $t \mapsto (x,y,z)$ given by
$$
x(t) = 2t \quad ; \quad y(t) = 3t \quad ; \quad z(t) =1
$$
The vector-valued version is just tidier and more convenient sometimes. It's the same reason we write vectors in general, rather than just individaul coordinates.
