(Solved): 6. In a tetrahedron ABCD let S be the midpoint of AD,R be the midpoint of BC and P be a point on AC ...

6. In a tetrahedron $\mathrm{ABCD}$ let $\mathrm{S}$ be the midpoint of $\mathrm{AD},\mathrm{R}$ be the midpoint of $\mathrm{BC}$ and $\mathrm{P}$ be a point on $\mathrm{AC}$ such that $\frac{AP}{PC}=\frac{1}{2}$. Denote $\vec{v}=\overrightarrow{AB},\vec{w}=\overrightarrow{AC}$ and $\vec{q}=\overrightarrow{AD}$. Present the following vectors as linear combinations of $\vec{v}$, $\vec{w}$ and $\vec{q}$ : $\overrightarrow{AP},\overrightarrow{AR},\overrightarrow{PR},\overrightarrow{PS},\overrightarrow{RD}$ 7. In the notations of the previous problem, let $T$ be the point of intersection of the plane PRS with the edge BD. Find $\frac{BT}{BD}$.