这几个都是动词和名词两种形态。
Cooperate表示为了达成某事与某人"合作"或者在某事上"协助配合"某人
Collaborate 意思和用法范围较小，多用于书籍或者研究上的“合作，协作”，另外还有“通敌”的意思
Coordinate强调协调，比如肢体协调，跟前两个差别较大

1. cooperate / cooperation

vi. 合作；协力；配合
n. 合作；协作；合作社；协同作用
http://dict.cn/cooperate

I am so happy to cooperate with you.
很高兴与你合作。

I advise you not to cooperate with that deceitful businessman.
我建议你不要和那个不诚实的商人合作。

Thanks for your cooperation.
多谢合作。

Our success was attributed not so much to your help as to our cooperation.
我们的成功与其归功于您的帮忙，不如归功于我们的合作。

2. collaborate / collaboration

vi. 合作；通敌
n. 合作；通敌
http://dict.cn/collaborate

She collaborated with her sister/She and her sister collaborated on a biography of their father.
她和姐姐合作写父亲的传记。

We would ask you to collaborate with us.
我们愿意请你们和我们合作。

This supplier and I collaborate on certain points.
我与供应商有数方面或以上的合作。

Then the long collaboration fell apart.
长期的兄弟合作就这样破裂。

He wrote the book in collaboration with his colleagues.
他与同事合作写了这本书。

It was a collaboration that produced extremely useful results.
这是一次富有成果的合作。

He was shot for collaboration with the enemy.
他因通敌而被枪毙了。

They were going to be punished for collaboration with the enemy.
他们将会因为通敌而受到惩罚。

3. coordinate / coordination

v. （使）协调；（使）一致；（使）同等
n. 协调

1. You must coordinate the movements of your arms and legs when swimming.
   游泳时你必须使臂和腿的动作相协调。
2. If we coordinate our efforts we should be able to win the game.
   如果我们同心协力,我们应该能够打赢这场比赛。
3. You must coordinate what you said with what you did.
   你必须使你的言行一致。

These dancers have poor coordination.
这些舞蹈演员协调能力不好。

Learning to drive is all a matter of coordination.
学开车主要是靠协调。

另外coordinate还有坐标的意思，用做名词

1. The filter is defined as a box of coordinates.
   过滤器定义的是一个方框的坐标。

还可以用做形容词

adj. 同等的；等位的；(大学)男女分院制的

1. This coordinate system is called the triad at the point.
   这个坐标系叫做该点的三面形坐标系。