Find change of direction given three points

Question

There are three points with $(x,y)$ coordinates, $A=(x_1, y_1), B=(x_2, y_2), C=(x_3, y_3)$. I want to find out the change of direction if you follow a straight line from $A$ to $B$ and then from $B$ to $C$, which will be the $\theta$ angle in the figure.

The magnitude of $\theta$ is easy to compute using dot product. If we define $\mathbf{v}=(x_2-x_1, y_2-y_1)$ and $\mathbf{w}=(x_3-x_2, y_3-y_2)$, then $\theta=\arccos \frac{\mathbf{v}\cdot \mathbf{w}}{\lVert \mathbf{v}\rVert \lVert \mathbf{w}\rVert}$.

Here is my question. I am also interested in whether $\overrightarrow{BC}$ lies to the left of direction of $\overrightarrow{AB}$ (as shown in the figure) or to the right. How do I get that in a simple way?

Answer 1

You can lift the situation from $\mathbb R^2$ to $\mathbb R^3$ by setting $\vec a = \begin{pmatrix} \vec v \\0 \end{pmatrix}$ and $\vec b = \begin{pmatrix} \vec w \\0 \end{pmatrix}$, and then using the cross product:

If $\vec a \times \vec b$ has positive $z$ coordinate, you get one direction, if it has a negative $z$ coordinate you get the other one.

An alternative aproach (but when it comes to the calculation it is basically identical) is first calculating a perpendicular vector $\vec v^\perp := \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}\vec v$ and then seeing wheter the scalar product $\vec v^\perp \cdot \vec w$ is greater or less than zero.

Answer 2

Let $\mathbf{v}=(v_1, v_2)$ and $\mathbf{w}=(w_1, w_2)$ be the vectors. The sine of the angle $\theta$ going from $\mathbf{v}$ to $\mathbf{w}$ is proportional to the following determinant:

$$D=\begin{vmatrix} v_1 && v_2 \\ w_1 && w_2 \end{vmatrix}$$

If $D>0$ then you are turning left; otherwise, you are turning right.

Note that this is simply the relevant computation when lifting both vectors to $\mathbb{R}^3$ and using the cross product.