Relative motion

In the derivation below, $\hat{\imath}$ is a unit-vector toward the East, and $\hat{\jmath}$ is a unit-vector toward the North. We want to find the velocity of $A$ relative to $C$ .

$\large{v_A - v_B = v_{AB} = \alpha (\hat{\imath} + \hat{\jmath}) \\ v_B - v_C = v_{BC} = \alpha (-\hat{\imath} + \hat{\jmath}) \\ v_A - v_C = v_{AB} + v_{BC}= \alpha (\hat{\imath} + \hat{\jmath}) + \alpha (-\hat{\imath} + \hat{\jmath}) = 2 \alpha \hat{\jmath}}$

This corresponds to "North".