Lemonade problem

O

oratonastick

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Joined
Jun 23, 2020
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So, here's my problem:

I'm working for a company that produces lemonade syrups. I've been testing new recipes using 4 different ingredients. I've identified 4 lemonade recipes containing these 4 ingredients that taste well. Now my boss asks me to create 2 syrups that can be combined in different ratios to mix each of these 4 lemonade recipes. Can this be done?
 
First of all, they taste "good" not "well". As far as the mathematics is concerned whether that "can be done" depends on the proportions of the ingredients.
 
If you happen to be taking a linear algebra course, you might recognize that you have four points in a four-dimensional space, and you want to find a basis consisting of two vectors. Under what conditions might that be done?
 
@Dr.Peterson: Thank you, that is helpful! I had to dig pretty deep (it been a while since I took any kind of math course), but I think I understand:
A combination of two vectors can be used to define a plane. For any 3 points in a 3-, 4-, n-dimensional space a plane can be defined that has all 3 points on it. Only if the 4th point happens to be on this same plane, all 4 points can be defined as a combination of 2 vectors. So if the 4 points are not on a plane, there is no solution to my problem. If they are on the same plane, I guess an infinite number of solutions is possible. Is this correct?
 
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