homework and exercises - Problem understanding a step in derivation of Lorentz Transformation

I need to understand a step in the derivation of Lorentz Transformation.

I cannot understand how can we equate the equations 1 and 2. I am talking about the relation between ct, x, y, z. Where did it come from?

Answer

In the Euclidean space $R^3$ you know that $$ r^2 = x^2 + y^2 + z^2 $$ Add to this the fact that $$ r = c \cdot t $$ $$ \text{(space = velocity} \cdot \text{time)} $$ and you can easily obtain the relation between $x$, $y$, $z$ and $c \cdot t$: $$ r^2 = x^2 + y^2 + z^2 = (c \cdot t)^2 $$ (do the same thing for the primed system)