If an arrow initially pointing North is rotated $450^{\circ }$ anticlockwise, which direction will it point to in its final orientation?

Let's simplify the large angle of rotation by eliminating full circular rotations ($360^{\circ }$ per rotation):
$$\text{Net Rotation}=450^{\circ }(\mathrm{mod}360^{\circ })$$

$$450^{\circ }=360^{\circ }+90^{\circ }$$

Rotating an object by $450^{\circ }$ anticlockwise is identical to rotating it by a single full turn ($360^{\circ }$) plus a net additional $90^{\circ }$ anticlockwise rotation.

1. Initial orientation: North.
2. Full $360^{\circ }$ turn anticlockwise brings it back to North.
3. Adding the remaining $90^{\circ }$ anticlockwise shift turns the arrow from North toward the left, which points directly West.