You are on your way to work. The road you usually take is under construction and to avoid congestion, you end up taking another route to work. You walk in late.
When using alternative as an adjective, you say: “I had to take an alternative route to work.” It implies the route you took was “one of a number of possible choices”.
When using it as a noun, you’ll say: “I had no alternative but to take the longer route.”
Alternatively is an adverb. It is used in places where there is need for “instead, on the other hand”. For example: “The usual route to work will see a lot of traffic congestion due to construction work. Alternatively, take the longer route.”
Alternate is used to mean substitute or stand-in. For example, “Since Prabhakar is not available today, Arun is the alternate trainee.”
Alternate is also used as a verb. Alternate as a verb is used to convey an action or an event that is interchanging repeatedly. For example, if you are taking care of someone while they are ill, you alternate between cold and hot compresses.
The adverb form of alternate is alternately which means “taking turns/succeeding by turns”. Your company has been handed a big project. Your boss is dilly dallying over the department he wants to hand over the project to. When he finally makes his decision, this is the sentence you can use- “He chose people from two departments alternately for this project.”