1.1.44 Vector Algebra

\ (\ underline {\ mathrm {y} _ {\ mathrm {c} \ text {}}}} \) on vectors \ (\ bar {a} = \ {1; 1; 3 \}, \ Bar} = {1; 2; 2 \ \}, 2 \}, \}, \ bar {c} = \ {0; Calculate: a) a scalar product of vectors \ (\ bar {a}, \ bar {b} \), their lengths, as well as the cosine of the angle between them. b) the vector work of vectors \ (\ bar {a}, \ bar {b} \), checking its perpendicularity to each of the dubbing with the help of a scalar work, the area of the corner formed by them and the sinus of the angle between them. Check the latter with the help of a basic trigonometric identity. c) the mixed work of vectors \ (\ bar {a}, \ bar {b}, \ bar {c} \) through the determinant, checking it using the vector work found above, as well as the volume of the parallelepiped and its height, lowered on the plane of the verge of vectors \ (\ bar {a}, \ bar {b} \).