1.1.20 Vector Algebra

\( \underline{\text { condition: }} \) Given the coordinates of the vertices of the pyramid \( A B C D \). Required: 1) write the vectors \( \overrightarrow{A B}, \overrightarrow{A C}, \overrightarrow{A D} \) in the orthonormal basis \( \vec{\imath}, \vec{\jmath}, \vec{k} \) 2) find the modules of these vectors; 3) calculate the scalar product \( (\overrightarrow{A B}+\overrightarrow{A C}) \cdot \overrightarrow{A D} \); 4) calculate the vector product \( (\overrightarrow{A B}-\overrightarrow{A C}) \times \overrightarrow{A D} \). \( A(3 ; 3 ;-3), B(7 ; 7 ;-5), C(5 ; 14 ;-13), \quad D(3 ; 5 ;-2) \).