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Suppose I know the following 3 normalized vectors in 3-D Euclidean space:

$\frac{\vec{A}}{||\vec{A}||}$,

$\frac{ \vec{A} + \vec{B}}{||\vec{A} + \vec{B}||}$

$\frac{ \vec{A} + \vec{C}}{||\vec{A} + \vec{C}||}$

Is it possible to determine the 3 unknown vectors $\vec{A}, \vec{B}, \vec{C}$ ?

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2 Answers

Hint: Suppose that from $\vec{A}, \vec{B}, \vec{C}$ you get the three normalized vectors $$\vec{D} = \frac{\vec{A}}{\|\vec{A}\|}, \quad \vec{E} = \frac{\vec{A} + \vec{B}}{\|\vec{A} + \vec{B}\|}, \quad \vec{F} = \frac{\vec{A} + \vec{C}}{\|\vec{A} + \vec{C}\|}.$$ Which vectors would you get if, instead of $\vec{A}, \vec{B}, \vec{C}$, you started with the vectors $2\vec{A}, 2\vec{B}, 2\vec{C}$?

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Take $A=(\alpha, 0)$, $B=(0, \alpha)$, $C=(-\beta,0)$ with $ \beta>\alpha >0$. Your three vectors end up the same for all values of the parameters.

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