Let \( A ( 4, 3, 5 ) \) and \( B ( 2, 6 , 0) \) be two fixed points. Find the position vector of the point which lies on the line segment \( \overline{AB} \) and internally divides it in the ratio \( 7 : 8. \)

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The above rectangular cuboid has base dimensions \(a=12\) and \(b=8.\) If the centroid \(I\) of triangle \(ACF\) internally divides the diagonal \(\overline{BH}\) of the rectangular cuboid in the ratio \(BI:IH=1:m,\) what is the value of \(m?\)

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Suppose three vertices of a parallelogram are \[O=(0,0,0), A=(1,5,6), B=(-1,-5,5),\] where \(\overline{OA}\) and \(\overline{OB}\) are adjacent sides. If \(C\) is the final vertex of the parallelogram, then what are the coordinates of \(C ?\)

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Consider a triangle \(ABC\) such that the centroid of of the triangle is \(G=(5,-2,3)\) and the midpoint of \(\overline{AB}\) is \(P=\left(1,\frac{1}{2},\frac{1}{2}\right).\) What are the coordinates of the point \(C ?\)

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