1.1.4 Vector Algebra

Problem: Given vectors $\overrightarrow{a}=\overrightarrow{OA},\overrightarrow{b}=\overrightarrow{OB},\overrightarrow{c}=\overrightarrow{OC},\overrightarrow{d}=\overrightarrow{OD}$. Beams $OA,OB$ and $OC$ are edges of e trihedral angle $T$. 1) Prove that the vectors $\overrightarrow{a},\overrightarrow{b},\overrightarrow{c}$ are linearly independent. 2) Decompose vector $\overrightarrow{d}$ into vectors $\overrightarrow{a},\overrightarrow{b},\overrightarrow{c}$ (solve the resulting system of equations using the inverse matrix). 3) Determine whether point $D$ lies inside $T$, outside $T$, on one of the boundaries of $T$ (which one?). 4) Determine for what values of the real parameter $\lambda$ the vector $\overrightarrow{d}+\lambda \overrightarrow{a}$, plotted from the point $O$, lies inside the trihedral angle $T$. $$\overrightarrow{a}=\overrightarrow{\{3;2;1\}},\overrightarrow{b}=\overrightarrow{\{1;-1;-2\}},\overrightarrow{c}=\overrightarrow{\{-2;3;5\}},\overrightarrow{d}=\overrightarrow{\{7;4;1}.$$