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Having two vectors $a$ and $b$ and with $b$ fixed, how would I go about increasing the angle between the two vectors by a specified $\Delta \theta$?

The dot product equation:

$$ a \cdot b = |a| \cdot |b| \ \cos \theta .$$

gives multiple solutions in the form of a cone for $3$-dimensional vectors. I want the new vector $a'$ to be in the same plane spanned by $a$ and $b$.

Also can this case be extended to arbitrary number of dimensions?

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    I feel like your question is a bit ambiguous. Increase the angle subject to a constraint? Without any constraint, as you mentioned, in $\mathbb R ^3$ this yields multiple solutions, just like in $\mathbb R ^2$ it would and certainly in higher dimensions as well.2011-11-09
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    Ah, the constraint is that it must be in the plane spanned by $a$ and $b$.2011-11-09

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