A man facing South - East makes three turns of 90 degrees in the anticlockwise direction and then two 45 degree turns in the clockwise direction. In which direction is his right hand now with respect to him?

Let's track the angular changes:
* Initial direction: South-East ($315^{\circ }$ or $-45^{\circ }$)
* Anticlockwise turns: $+3\times 90^{\circ }=+270^{\circ }$
* Clockwise turns: $-2\times 45^{\circ }=-90^{\circ }$
* Net turn $=+270^{\circ }-90^{\circ }=+180^{\circ }$ (Anticlockwise)

Facing exactly opposite to South-East means the man is now facing North-West.
If a person is facing North-West, their right hand points $90^{\circ }$ clockwise from North-West, which is North-East.