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Question

# Solve using Lagrange multipliers. Find a vector in 3-space whose length is 5 and whose components have the largest possible sum.

Solution

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Let $\textbf{v}=\langle x,y,z\rangle$ be a vector in $\mathcal{R}^3$ with lenght $||\textbf{v}||=5$. This is

\begin{aligned} \sqrt{x^2+y^2+z^2}=5\equiv x^2+y^2+z^2=25 \end{aligned}

Let $f(x,y,z)=x+y+z$ the sum of the vector components. We need to maximize $f$ under the constraint $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 25$. Using Lagrange Multipliers

\begin{aligned} \nabla f=\lambda \nabla g\Rightarrow \langle 1,1,1 \rangle = \lambda \langle 2 x , 2 y , 2 z \rangle\\ \lambda = \frac { 1 } { 2 x } = \frac { 1 } { 2 y } = \frac { 1 } { 2 z } \Rightarrow x = y = z \end{aligned}

Using this in the given constraint:

\begin{aligned} x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 25 \Rightarrow 3 x ^ { 2 } = 25 \Rightarrow x = y=z=\pm \frac { 5 \sqrt { 3 } } { 3 } \end{aligned}

We note that for $x=y=z=\frac{5\sqrt{3}}{3}$, we obtain the largest component sum

\begin{aligned} S=3\frac{5\sqrt{3}}{3}=5\sqrt{3} \end{aligned}

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