What is the maximum possible length of the vector resulting from the following linear combination? $$ \frac{1}{\| \mathbf{v_1} \|} \,\mathbf{v_1} + \frac{1}{\|\mathbf{v_2}\|} \,\mathbf{v_2} + \cdots + \frac{1}{\| \mathbf{v_n} \|} \,\mathbf{v_n} $$
2026-04-30 00:08:22.1777507702
Precalculus linear combination problem
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Let $\mathbf{u_i} = \frac{1}{\| \mathbf{v_i} \|} \,\mathbf{v_i}$, then $\| \mathbf{u_i} \| = 1.$
$ \Big\|\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}\Big\| = \| \mathbf{u_1} + \mathbf{u_2} + \dots + \mathbf{u_n}\| \le \| \mathbf{u_1}\| + \|\mathbf{u_2}\| + \dots + \|\mathbf{u_n}\| = n.$