Solution

 (a) 

$T_{s}=T\cdot a_{s}=\frac{T\cdot S}{|S|}=\frac{(2,-6,-3)\cdot (1,2,1)}{\sqrt{6}}=\frac{-7}{\sqrt{6}}=-2.8577$

(b) $S_{T}=(S\cdot a_{T})a_{T}=\frac{(S\cdot T)T}{T^{2}}=\frac{-7(2,-6,3)}{7^{2}}\\\\\\=-0\cdot 2857a_{x}+0\cdot 8571a_{y}-0\cdot 4286a_{z}$

(c)

$$sin\:\theta_{TS}=\frac{|T\times{S}|}{|T||S|}=\begin{vmatrix}2&-6&3\\1&2&1\end{vmatrix}=\frac{|(-12,1,10)|}{7\sqrt{6}}=\frac{\sqrt{245}}{7\sqrt{6}}=0\cdot 9129\\\\\\\theta_{TS}=65. 9^{\circ}$$