Point Parallel Form Describe Same Line as Point Normal Form

Question

And that's how far I able to get, any suggestion how I can equate both (bold) equation or did I do totally wrong?

Answer 1

You're pretty much almost there.

Remember that we are working with vectors, so we can equate their components. So, the first equation $(x,y) = (p_1,p_2) + t (u_1, u_2)$ can be expressed in a system of two equations. $$\begin{align} x &= p_1 + t \cdot u_1 \tag1 \\\ y &= p_2 + t\cdot u_2 \tag2 \end{align}$$

This system describes a line. However, we can describe the same line by getting rid of that parameter $t$. We do that by solving for $t$ in $(1)$, and then substituting it into $(2)$. We will find that $$t = \frac{1}{u_1}(x-p_1).$$

Putting that into $(2)$, and after some algebraic manipulation, we have $$\begin{align} y &= p_2 + \frac{u_2}{u_1}(x-p_1) \\ y-p_2 &= \frac{u_2}{u_1}(x-p_1) \\ u_1(y-p_2) &= u_2(x-p_1).\end{align}$$

You should notice that this last line is the same as your last line. Hence, both equations describe the same line.