Dalia S.

# Among all the unit vectors find the one for which the sum is minimal

Among all the unit vectors u= <x,y,z> in R3,
find the one for which the sum x + 2y + 5z is minimal.

u=<>?

## 2 Answers By Expert Tutors

By:

Tutor
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Math and Science Tutor with an Engineering Background

Tutor
4.9 (135)

Patient MIT Grad For Math and Science Tutoring

Byron S.

A unit vector doesn't have to lie on an axis, it just has to have a magnitude of 1.
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10/29/14

Russ P.

I agree.  Your three unit vectors just need to be independent so they collectively span every point in R3 space.  But usually, they are taken as orthogonal to simplify calculations.  I thought of your interpretation of the problem as well (which is more general than mine) but decided that Dalia only works with orthogonal unit vectors placed along the x, y, and z axes, so why complicate the solution?

Thanks to your solution, Dalia is now aware that people can interpret problems differently
(based on their own assumptions to fill in holes in the problem statement) and different solutions can result.
Russ
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10/29/14

Dalia S.

the answers do not seem to be correct
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10/30/14

Russ P.

Dalia,
Can you tell us what the correct answers are, so we can figure out what the problem writer was aiming for?
Russ
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10/30/14

Dalia S.

im not too sure if i am doing something wrong
u=?
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10/30/14

Russ P.

Dalia,
Yes, that works and agrees with Byron's interpretation of the problem and solution.
Thanks, Russ
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10/30/14

Dalia S.

that answer seems to be incorrect, is it possible that a mistake might have been made?
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10/31/14

Russ P.

Dalia,

What is troubling you about it?  Please be more specific. In the meantime, let me take another crack to reassure you that the answer is correct.

The problem asked to find that unit vector u that minimized the scalar function f(x,y,z) = (x + 2y + 5z).  So you plug in the (x,y,z) components of u which are just scalar numbers into the x,y and z in the f(x,y,z), work through the arithmetic, and get f = -√30 as its scalar value.  This agrees with Byron's (x+2y+5z) = (√30) cos Θ when Θ=180o and its cosine is minus 1.  Maybe you're not comfortable with his dot product approach to get the solution?

To summarize, u is the unit vector, f is not a vector, just a scalar function which could take on many values, but has its minimum value when the unit vector points in a certain direction as given by Byron's answer.  For example, my interpretation of the problem added a constraint (to consider only the unit vectors along the orthogonal x, y & z axes) and produced an f value of +1, which is bigger than Byron's -√30.  So his interpretation and solution did a better job of minimizing the scalar f without imposing my constraint to stay with the orthogonal x,y & z unit vectors.  Hence, his approach is the more general and preferred one.

Russ
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10/31/14

Byron S.

You may be expected to rationalize the denominators of the solution.

Alternately, if Russ' approach seems to fit better, you might try the negative xyz basis vectors 〈-1, 0, 0〉, 〈0, -1, 0〉, 〈0, 0, -1〉. The -z vector gives a result of -5, which is the smallest of those, but still not quite as small as -√(30).
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10/31/14

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