Answer:
[tex]\vec B = \left(x_{A}+\frac{5}{6}\cdot (x_{C}-x_{A}), y_{A}+\frac{5}{6}\cdot (y_{C}-y_{A}), z_{A}+\frac{5}{6}\cdot (z_{C}-z_{A})\right)[/tex]
Step-by-step explanation:
Let suppose that A, B, and C have the following points with respect to origin in the Euclidean space:
[tex]\vec A = (x_{A}, y_{A}, z_{A})[/tex]
[tex]\vec B = (x_{B}, y_{B}, z_{B})[/tex]
[tex]\vec C = (x_{C}, y_{C}, z_{C})[/tex]
Besides, let consider that locations of A and B are currently known. The ratio is:
[tex]\frac{AB}{AC} = \frac{5}{6}[/tex]
[tex]AB = \frac{5}{6} \cdot AC[/tex]
Vectorially speaking, expression can be rewritten in the following terms:
[tex]\overrightarrow{AB} = \frac{5}{6}\cdot \overrightarrow{AC}[/tex]
[tex](x_{B}-x_{A}, y_{B}-y_{A}, z_{B}-z_{A}) = \frac{5}{6}\cdot (x_{C}-x_{A}, y_{C}-y_{A}, z_{C}-z_{A})[/tex]
Now, each side of the equation is summed vectorially by [tex]\vec A[/tex] and coordinates of point B are finally found:
[tex]\vec B = \left(x_{A}+\frac{5}{6}\cdot (x_{C}-x_{A}), y_{A}+\frac{5}{6}\cdot (y_{C}-y_{A}), z_{A}+\frac{5}{6}\cdot (z_{C}-z_{A})\right)[/tex]
