B

Find the area of the triangle \(ABS\). Only first two coordinates of the point $B=[2;0;?]$ are given and $B$ lies in the plane $\alpha$ defined by the equation \[ \alpha \colon 2x + y - z - 5 = 0. \] The point \(S\) is the intersection point of the plane \(\alpha \) and the line \(k\) which is perpendicular to \(\alpha \) and passes through the point \(A = [0;0;1]\).

Find the value of the parameter \(a\) which ensures that the vectors \(\vec{u} = (3;a;-2)\) and \(\vec{v} = (-6;4;a - 3)\) are perpendicular.

Given points \(A = [1;1]\), \(B = [5;2]\) and \(C = [8;7]\), find the angle \(\measuredangle ABC\).

Consider a parallelogram \(ABCD\) with \(A = [1;3]\), \(B = [2;-1]\) and \(C = [5;1]\). Let \(S\) be the center of the diagonal \(BD\). Find the vector \(\overrightarrow{AS } \).