This one is a little tricky! "Northerly" is an adjective that refers to a vague direction. Imagine that you are facing north, and waving your hand around to suggest a wide range of directions from WNW to ENE. All of those directions are "northerly". You can think "from somewhere in the vague vicinity of north". It may not be directly north. "Northern" is an adjective, meaning a region that is north of you. Maine is a northern state in the US" for example. "The Canadian Goose is a northern bird." Northern is NOT a noun, and cannot be used as one. "North" is a noun. It is the direction, and it is the exact direction. It does not refer to a region. But here's where things get tricky... "North" is ALSO an adjective! So in many places that you can use "northern", you can also use "north"! Not all situations, but your example sentences are some places where this works. "We set off in a northerly direction." This is correct, but remember, this might not mean exactly north. The trick here are the words "in a". Because "northerly" covers a wide range, you can be "in a northerly direction", because this is a wide area. You can't be "in a northern direction" because there is only one northern direction. You can, however, say "We set off north." "The northern end of a bridge" is the most correct answer, but strangely enough, you can also say "The north end of a bridge." Both are correct. You cannot say "The northerly end of a bridge" because it makes no sense to be vague when there are only two ends of a bridge. "A northern wind" and "A north wind" are both correct, as I mentioned above.

Most of the time, you need to use "north" and, of course, the same goes for the equivalent west/south/east. North can be used as a noun, as in: "Mount Ranier lies to the north." It can be used as an adjective: "I'll meet you at the north gate." And it can be used as an adverb: "We are sailing north tomorrow." So it is very versatile and can be applied in most situations when you need to use a direction. Northern is an adjective. It is only used as an adjective, but it can be used interchangeably with 'north' whenever north would have been used as the adjective in a sentence. So I could say: "I'll meet you at the northern gate." But in my other two examples above, "northern" can't be substituted for north. "Northerly" is kind of confusing. Usually, northerly would be used for wind, storms, other weather patterns and it doesn't mean that the wind/storm is moving north like you might expect. It actually means that the wind/storm is coming from the north and is moving south/southeast/southwest. So you don't use northerly in a sentence like your example. If you wanted to use it to describe the movement of people, you would need to say, "They came from a northerly direction," and it changes the meaning of the sentence to mean that they are moving south. So, in your example sentences: "We set off in a northern direction." Or, you could say, "We set off going north." "The northern end of the bridge." Or, "the north end of the bridge." "A northerly wind was blowing last night." (But, this means the wind is coming from the north, blowing south.)

We set off in a northerly direction. - This is a general direction, not necessarily true north but what the user estimates as being north The northern end of a bridge - this is describing a place A north wind was blowing hard last night. - this is more the direction the wind was blowing from I hope that helps a bit.