Given three points, is the angle positive or negative?

Question

Suppose we have the points A, B, and C on a plane. Lets say we have the line AB and AC. Is there a way to figure out if the angle between BAC is positive or negative?

Ie. did we have to do a positive rotation to get the line AB to AC (using A is the pivot) for some theta or a negative rotation?

Answer 1

Hint, the relation between the distances $AB$, $AC$ and $BC$ and angle $BAC$ is given by

$$ |BC|^2 = |AB|^2 + |AC|^2 - 2 |AB| |AC| \cos\Big(BAC\Big) $$

Answer 2

Hint: Use the cross product $ \mathbf{a} \times \mathbf{b} = \left\| \mathbf{a} \right\| \left\| \mathbf{b} \right\| \sin \theta \ \mathbf{n} $

$\mathbf{n}$ is a unit vector perpendicular to the plane containing a and b in the direction given by the right-hand rule