(c) Consider any two vectors u and v in the plane. (i) By considering the dot product, prove that: lullv| zu.v (ii) By squaring both sides of the inequality, prove that: 121+ 10/2 12+ 01 (iii) By plotting the vectors u, v and u + v, explain why the above inequality must be true.(e) Suppose that u and v are two vectors in the plane and let u = Proj(u) + we the asson vector decomposition. (i) Explain why: (171 1212 - 12. 0/2 10/2 + 1 w / 2 (ii) Hence deduce that for any vectors u, v: la . n/ z lallal And determine when equality holds. (18](b) Consider the two vectors ( my ) and (ma ) (i) Explain why their gradients are m, and m2 respectively. [13] (ii) When is the dot product of these two vectors equal to 0? What is the significance of this result? [14]