1.1.44 Vector Algebra

\( \underline{\mathrm{y}_{\mathrm{c} \text { catch: }}} \) On vectors \( \bar{a}=\{1 ; 1 ; 3\}, \bar{b}=\{1 ; 2 ; 2\}, \bar{c}=\{0 ; 2 ; 1\} \) a parallelepiped is constructed. Calculate: a) The scalar product of the vectors \( \bar{a}, \bar{b} \), their lengths, as well as the cosine of the angle between them. b) The vector product of vectors \( \bar{a}, \bar{b} \), checking its perpendicularity to each of the factors using the scalar product, the area of ​​the face they form and the sine of the angle between them. The latter can be verified using the basic trigonometric identity. c) The mixed product of the vectors \( \bar{a}, \bar{b}, \bar{c} \) through the determinant, checking it using the vector product found above, as well as the volume of the parallelepiped and its height lowered onto the plane of the face of the vectors \( \bar{a}, \bar{b} \).