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

In summary, the conversation discusses a linear algebra problem where the goal is to find the minimum value of a2 + b2 + c2 given a constraint of scoring 76% in the course. The problem involves finding a point on a plane that has the smallest norm, and the conversation participants are discussing different approaches to solving it.
  • #1
djh101
160
5

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.
 
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  • #2
So you need to an (a,b,c) on 0.2a+0.3b+0.5c=76 such that [itex]\|(a,b,c)\|[/itex] is minimal.

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

Does this give you an idea?
 
  • #3
That helps a little. I'm very tired, though, so I'll see if I can figure it out tomorrow. Thank you.
 
  • #4
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?
 
  • #5
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.
 

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

1. What is an orthogonal projection?

An orthogonal projection is a mathematical technique used to find the closest point on a given line or plane to a given point in space. It involves finding the shortest distance between the point and the line or plane, and then projecting the point onto that line or plane.

2. How is the formula for minimizing a^2 + b^2 + c^2 derived?

The formula for minimizing a^2 + b^2 + c^2 is derived from the Pythagorean theorem, which states that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. In this case, the hypotenuse represents the distance between the given point and the line or plane, while the other two sides represent the values of a, b, and c in the formula.

3. What is the purpose of minimizing a^2 + b^2 + c^2 in orthogonal projections?

The purpose of minimizing a^2 + b^2 + c^2 is to find the point on a given line or plane that is closest to a given point in space. This can be useful in various applications such as optimization problems, computer graphics, and data analysis.

4. How is the formula for minimizing a^2 + b^2 + c^2 applied in real-world scenarios?

The formula for minimizing a^2 + b^2 + c^2 can be applied in various real-world scenarios, such as finding the shortest distance between a point and a line in a 3D space, optimizing the placement of objects in a 2D or 3D environment, and minimizing error in statistical data analysis.

5. Are there any limitations to using orthogonal projections to minimize a^2 + b^2 + c^2?

One limitation of using orthogonal projections to minimize a^2 + b^2 + c^2 is that it assumes a linear relationship between the given point and the line or plane. In cases where the relationship is non-linear, this method may not provide an accurate solution. Additionally, this method may not work in cases where the given point is not within the range of the line or plane.

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