Orthogonal Projections: Minimize a^2 + b^2 + c^2

[djh101]

Homework Statement

There are three exams in your linear algebra class and you theorize that your score in each exam will be numerically equal to the number of hours you study. The three exams count 20%, 30%, and 50% and your goal is to score 76% in the course. How many hours, a, b, and c should you study for each exam to minimize a² + b² + c²?

Homework Equations

.2a + .3b + .5c = 76
a² + b² + c²

The Attempt at a Solution

I'm not really sure to begin. I assume this has something to do with orthogonal projections, since it is in the orthogonal projections chapter/section, but the section doesn't really go over anything related to this question (my book is by Bretscher and he seems to have a habit of doing this). All I really need is a kick start.

 

[micromass]

So you need to an (a,b,c) on 0.2a+0.3b+0.5c=76 such that $\|(a,b,c)\|$ is minimal.

So you need to find a point on a plane such that its norm is minimal.

Does this give you an idea?

 

[djh101]

That helps a little. I'm very tired, though, so I'll see if I can figure it out tomorrow. Thank you.

 

[craned]

Hi, I'm stuck on this problem too and can't seem to figure it out even with your hint. Could you explain it a bit more explicitly?

 

[djh101]

Yeah, I'm still a little stuck. I have all the pieces and know what they are geometrically, but I'm still a little lost on what to do with them.