1.1.4 Vector Algebra

Problem: Given vectors \( \vec{a}=\overrightarrow{O A}, \vec{b}=\overrightarrow{O B}, \vec{c}=\overrightarrow{O C}, \vec{d}=\overrightarrow{O D} \). Beams \( O A, O B \) and \( O C \) are edges of e trihedral angle \( T \). 1) Prove that the vectors \( \vec{a}, \vec{b}, \vec{c} \) are linearly independent. 2) Decompose vector \( \vec{d} \) into vectors \( \vec{a}, \vec{b}, \vec{c} \) (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 \( \vec{d}+\lambda \vec{a} \), plotted from the point \( O \), lies inside the trihedral angle \( T \). \[ \vec{a}=\overrightarrow{\{3 ; 2 ; 1\}}, \quad \vec{b}=\overrightarrow{\{1 ;-1 ;-2\}}, \quad \vec{c}=\overrightarrow{\{-2 ; 3 ; 5\}}, \quad \vec{d}=\overrightarrow{\{7 ; 4 ; 1} . \]