1.1.71 Vector Algebra

condition: Given the vertices of the pyramid \( A B C D: A(2,3,1), B(4,1,-2), C(6,3,7) \), \( D(7,5,-3) \) 1) Construct a pyramid in the Cartesian coordinate system 2) determine the coordinates and lengths of the vectors \( \overline{A B}, \overline{A C}, \overline{A D} \) 3) determine the direction \( \overline{A B} \) (its direction cosines) 4) find \( (\overline{A B}, \overline{A C}) \) 5) find the projection of the vector \( \overline{A C} \) to the direction \( \overline{A B} \) 6) calculate \( [\overline{A B}, \overline{A C}] \) 7) calculate \( (\overline{A B}, \overline{A C})^{2} \) 8) write down the equation of the plane passing through the points \( A, B, C \) 9) find the area of the face \( A B C \) 10) find the volume of the pyramid \( A B C D \) 11) write the equation of the height lowered from the vertex \( D \) to the face \( A B C \) 12) find the length of the height, dropped from the point \( D \) to the face \( A B C \).