Say there are two unit vectors a and b. I want to find unit vector c such that the deviation between c and a is the opposite to the deviation between a and b. The angles between c and a, and a and b will be will be the same but in opposite directions and all three vectors will fit on a plane. How can I find c?
2026-04-06 20:39:09.1775507949
Find third vector with opposite deviation
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We want to find $c$ such that $a$ is exactly in-between $b,c$. To do so, we need to reflect $b$ about $a$. The easiest way is to project $b$ onto $a$, which is $(b \cdot a)a$ since $a$ is a unit vector. And then you just need to reverse the component $b - (b \cdot a)a$. So you get $(b \cdot a)a - (b - (b \cdot a)a) = 2 (b\cdot a)a - b$, which should make sense because it's exactly like reflecting in a mirror (2 times mirror position minus original position).