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# help plz

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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

### 2+0 Answers

#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
$$1+1+ \cdots + 1 = n$$

Feb 7, 2020
edited by heureka  Feb 7, 2020