How do I visualize if three points represent a right or left turn?

Question

Consider three points: $P_1$, $P_2$ and $P_3$. We have to decide whether $P_1P_2P_3$ represents a "right turn" (i.e. a turn in clockwise order) or a "left turn" (i.e. a turn in counter-clockwise order).

Here's a method to determine it.

For three points $P_1 = (x_1, y_1)$, $P_2 = (x_2, y_2)$ and $P_3 = (x_3, y_3)$, compute the z-coordinate of the cross-produt of the vectors $\overrightarrow{P_1P_2}$ and $\overrightarrow{P_1P_3}$, which is given by the expression $(x_2 - x_1)(y_3 - y_1) - (y_2 - y_1)(x_3 - x_1)$. If the result is $0$, the points $P_1$, $P_2$ and $P_3$ are collinear. If the result is positive, the three points constitute a "left turn" (or a counter-clockwise orientation), otherwise the points represent a "right turn" (or a clockwise orientation). This reasoning assumes counter-clockwise numbered points.

I am not interested in these calculations, rather I would like to be able to identify a "right" or "left" turn by looking an imagine containing three points $P_1$, $P_2$ and $P_3$. How do you usually interpret such pictures?

Answer 1

How about simply looking when approaching a point following the line, whether the next point is in the left half-plane or the right half-plane w.r.t the line (of your motion)?

Just as if you are driving a car, and your examples is a road map.

Answer 2

Assume $P_1 = (x_1, y_1), P_2 = (0,0), P_3 = (x_2,y_2)$. Whether this is a left or right turn is determined by looking at the sign of the determinant:

\begin{vmatrix} x_1 & y_1 \\ x_2 & y_2 \end{vmatrix}