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

Question

Among all the unit vectors u= <x,y,z> in R^3,

find the one for which the sum x + 2y + 5z is minimal. 

 

u=<>?

Answer 1

They key to this problem is recognizing that the sum you're given can be written as a dot product.

1x + 2y + 5z = u · v

Where u = 〈x, y, z〉 and v = 〈1, 2, 5〉

 

u · v = |u| |v| cos θ     where θ is the angle between the vectors

 

Since u is a unit vector, it has magnitude 1. v has magnitude √(1²+2²+5²) = √(30)

1x + 2y + 5z = |u| |v| cos θ

x + 2y + 5z = √(30) cos θ

 

This will be at a minimum when cos θ = -1, θ = 180º; u and v are in opposite directions.

-v = 〈-1, -2, -5〉

 

To make this a unit vector, divide by it's magnitude:

u = -v/|v| = 〈-1/√(30), -2/√(30), -5/√(30)〉

Answer 2

Dalia,

 

By definition a unit vector has magnitude 1 and lies along one of the three axes in R³.  Thus it has no components in the other two axes.

 

So in (x,y,z) format, unit vector x = (1,0,0) and the Sum, S = x + 2y + 5z = 1 + 2(0) + 5(0) = 1 +0+0 =1

So in (x,y,z) format, unit vector y = (0,1,0) and the Sum, S = x + 2y + 5z = 0 + 2(1) + 5(0) = 0 +2+0 =2

So in (x,y,z) format, unit vector z = (0,0,1) and the Sum, S = x + 2y + 5z = 0 + 2(0) + 5(1) = 0 +0+5 =5

 

Therefore, the sum S is minimal along the x unit vector.