Consider the directed network (D, w) with V(D) = {a, b, c, d, e}, and w: A(D)R with - A(D) {ab, ac, ace, bd, cd, ce, ea, eb}, w(ab) 2, w(ac)=2, w(ae) = 1, w(bd)=2, w(cd) 1, w(ce)=-3, w(ea)=1, w(eb)=1. Use the Bellman-Ford algorithm to find a shortest directed a-d-path in (D, w). Show your working, and give the path and its length.

Draw the network defined by N = {1, 2, 3, 4, 5, 6} A = {(1,2), (1,3), (1,4), (1,5), (2,3), (2,4), (2,5), (3,4), (3,6), (4,5), (4,6)}

Now consider the network (G, w) such that V(G) = {V₁, V2, V3, V4, V5, V6, V7}, E(G) = {V₁V2, V₁ V3, V2V3, V2V4, V2V5, V3V5, V3V6, V4 V5, V5V6} and w (v₁v₂): = 1, w(v₂v₁) = 2, w(v3v6) = 1, w(v₁v3) = 4, w(v₂v5) = 3, w(v4v5) = 1, w(v2V3) = 4 w (v3v5) = 4 w (v5 V6) = 3. Use Kruskal's algorithm to determine a minimum spanning tree of this network. Explain clearly what the algorithm is doing and draw the minimum spanning tree. Show that the spanning tree found in the previous part of the question is the unique minimum spanning tree of the network (G, w).

(2) Write v as a lincar combination of u₁, U₂, U3 if possible, where U₁ = (2,3,5), u₂ = (1,2,4), u3 = (-2,2,3) and v= = (10, 1,4)

4) Let a = (-3, 0, -4) and 6 = (-3, 3, 1). Find the projection of b onto a.

) Show that the points A(8,-1,0), B(0, 1, 1), and C(2, 1, -3) are non-collinear.

Question 2 Consider the directed network (D, w) with V (D) = {a, b, c, d, e}, A(D) = {ab, ac, ae, bd, cd, ce, ea, eb}, and w A(D) R with = w(ab) 2, w(ac) = 2, w(ae) = 1, w(bd) = 2, w(cd)=1, w(ce) = -3, w(ea) = 1, w(eb) = 1. (b) Use the Bellman-Ford algorithm to find a shortest directed a-d-path in (D, w). Show your working, and give the path and its length.

given u = <14,5> and v = <1,4>, find the projection of u onto v (a) <31, 13> (b) <28,8> (c) <5,11> (d) <2,8> (e) <28,10>

7. Find the centroid of a triangle whose vertices are (2, 3), (-4, 6) and (2, -6). B. (0, -1) A. (0, 1) C. (1, 0) D. (-1, 0)

In (R, 1sorg), if A = and B = (2, 3) then AUB =