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Given $n-1$ orthogonal unit length vectors $b_1, b_2, \ldots, b_{n-1}$ in the vector space $\mathbb{C}^n$, it's clear that there exsits a unique $b_n \in \mathbb{C}^n$ with unit length, which is orthogonal to all $b_1, \ldots, b_{n-1}$. In other words, $b_1, \ldots, b_{n-1}$ can be uniquely extended to an orthonormal basis.

Is there a simple explicit formula for $b_n$ in terms of $b_1, \ldots, b_{n-1}$?

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$b_n$ is not unique. It can be multiplied by a scalar of absolute value one. –  doug Mar 21 '13 at 21:25
You can take the Hodge star of $b_1 \wedge ... \wedge b_{n-1}$. Also, it's only unique up to multiplication by an element of the unit circle. –  Qiaochu Yuan Mar 21 '13 at 21:25

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