Signals that are plus or minus constants on unit intervals (here $\pm 2$) are great example, because they help you understand some of the concepts of orthogonality, like a combination of amplitude and support, and you can almost evaluate it visually, without much computations. Plus, you don't have to deal with complex conjugates (as Matt did).
First, orthogonality does not depend on amplitude, so you can divide your signals by $2$, to have $\pm 1$ amplitudes. With this, you'll only have to multiply $1 \times 1$, $-1 \times -1$ or $1 \times -1$, which save mistakes. On a graph paper, you can draw $s_1$, $s_2$ and their product:
Each square of the product is thus of "unit surface", either positive or negative:
- positive if $s_1$ and $s_2$ have the same sign;
- negative if $s_1$ and $s_2$ have opposite signs;
Signals will be orthogonal if there is the same number of positive and negative unit squares, so they sum up to zero surface. This is the layman version of the integral or sum definition of orthogonality for such simple signals, by counting areas multiplying support (unit $x$) and amplitude. So as there are three green bars up, and three down, they are orthogonal. Here is the same graph, showing same ($s$) and opposite ($o$) signs.
When you are used to this, to can do it more quickly, by cancelling unit squares of opposite signs, sequentially. For the first two $x$-units (say $1$ and $2$), $s_1$ and $s_2$ are of same-then-opposite signs, so they cancel each other. Then, $s_1$ and $s_2$ are of same signs on squares $5$ and $6$, but opposite on $3$ and $4$. Once again, those cancel each other.
- such sequences are important in mathematics and were in signal processing. They are related to (orthogonal) Walsh, Paley or Hadamard sequences (and Haar), that can be used as orthogonal transformations with only adds and subtracts, that can save time and power. One source: Transmission of Information by Orthogonal Functions, Harmuth, Henning F.
- This reasoning might seem quite simple, but it is the base of integration: cutting either the time or amplitude axes in chunks, and summing them.