Question 2 (35 marks)
(a) If \(\vec{u} = (2,0,0)\), \(\vec{v} = (4,k,1)\) and \(\vec{w} = (0,-4,8)\) are vectors in \(\mathbb{R}^3\), where k is a constant.
(i) Determine the possible values of k such that \(\vec{v}\) is orthogonal to \(\vec{p} = (0,R,0)\).
(Correct your answers to 4 decimal places).
[5 marks]
(ii) Determine the possible values of k such that the normal of the plane that contains \(\vec{u}\) and \(\vec{v}\) is parallel
to \(\vec{w}\).
[10 marks]
(b) Given the points, A(2,1,0), B(-1,3,7) and C(1,1,1).
(i) Find the two vectors \(\vec{u}\) and \(\vec{v}\) such that \(\vec{AB} = \vec{u} + \vec{v}\), where \(\vec{u}\) is parallel to \(\vec{BC}\) and \(\vec{v}\) is orthogonal
to \(\vec{BC}\). Correct your answers to 4 decimal places.
[10 marks]
(ii) Let D(0,-2,k) be another point in \(\mathbb{R}^3\). Find the value of k such that A, B, C and D lie on the same
[10 marks]
plane.
