A vigorous shake is a forceful or energetic shaking. For example, you might shake a cocktail shaker vigorously until the egg white is foamy. Link
Most of the drinks recommend we shake it before we drink, and sometimes when we do it doesn't seem to be enough. For example, I get carried away while shaking anything. So, I shake it like my life depends on it. It's like saying if you don't shake it very well, it'll kill you. So, you use your energy and shake it vigorously.
The point of it all is for the drink or whatever it is to mix up very well and circulate too. So, shaking vigorously is quite essential, which is why you need to shake vigorously for effectiveness.
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