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What is the maximum possible length of the vector resulting from the following linear combination? \(\frac{1}{\| \mathbf{v_1} \|} \,\mathbf{v_1} + \frac{1}{\|\mathbf{v_2}\|} \,\mathbf{v_2} + \cdots + \frac{1}{\| \mathbf{v_n} \|} \,\mathbf{v_n}\)
 

 Feb 7, 2020
 #1
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The maximum possible length is 1.

 Feb 7, 2020
 #2
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What is the maximum possible length of the vector resulting from the following linear combination?
\(\dfrac{1}{\| \mathbf{v_1} \|} \,\mathbf{v_1} + \dfrac{1}{\|\mathbf{v_2}\|} \,\mathbf{v_2} + \cdots + \dfrac{1}{\| \mathbf{v_n} \|} \,\mathbf{v_n}\)

 

I assume:


A unit vector is a vector of length 1.

The unit vector \(\hat{v}\) is defined by \(\dfrac{\mathbf{v}}{\| \mathbf{v} \|}\)

 

\(\begin{array}{|rcll|} \hline && \dfrac{1}{\| \mathbf{v_1} \|} \,\mathbf{v_1} + \dfrac{1}{\|\mathbf{v_2}\|} \,\mathbf{v_2} + \cdots + \dfrac{1}{\| \mathbf{v_n} \|} \,\mathbf{v_n} \\\\ &=& \hat{v} _1 + \hat{v} _2 + \cdots + \hat{v} _n \\ \hline \end{array} \)

 

 The maximum possible length of the vector resulting from the linear combination is
\( 1+1+ \cdots + 1 = n\)

 

laugh

 Feb 7, 2020
edited by heureka  Feb 7, 2020

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