Q3 Let u = (i) (ii) (iii) (iv) Q S T ν 'P -0. U R and w = 1 α. Calculate the angle between the two vectors u and v. Find the value of a such that u and w are orthogonal. Find the value of a such that u, v, w are linearly dependent. [5 marks] [5 marks] [15 marks] State the "Basis Theorem". Choose a value of a, different from the one that you have found in (iii), so that u, v and w become linearly independent, apply the "Basis Theorem" to obtain a basis for the three-dimensional space R3 using ´2 Gaussian Elimination Method. Find the coordinates of 3 in terms of this basis. 5 (Marks will ONLY be awarded for Gaussian Elimination Method.) [15 marks]

Fig: 1