please answer question 4-10

Q4. (10 pts) Calculate a . b, if la] = 5, |b| = 6, and la + b| = 9 Q5. (10 pts) For A = 2ye3x i + j and B = 3yex i + In (xy)j, show that: a(A . B) OB OA = A - . B ox ax Hint: A . B is a scalar, A, B are vectors. The vector derivative for A = Axi + Ayj : - dA dAx it dAy dx Q6. (10 pts) Calculate the gradient of f (x, y) = 7x8ey at (2,3). Hint: Vf (x, y) = of (x,y) ax it of ( x,y ) ay Q7. (10 pts) Calculate (a) the divergence and (b) the curl of A = sin(xy)i + In(xy)j at (2,3). Hint:V . A = Ax + 0Ay, V X A = dAy JAx k ax ay ax ay Q8. (10 pts) Calculate the Laplacian of f (x, y) = xy8 + 2xsin(3y): Hint: Af (x, y) = v2f (x,y) = of(xy) a2f(x,y) ax2 ay2 Q9. (10 pts) Evaluate S F . dr along the parametric curve x = 2u, y = Vu from u = 0 to u = 1 if F = -yi+ x2j. Q10. (10 pts) Evaluate I = SS z x2dS, where S is the plane defined by x + y + z? = 1, x > 0, y > 0 and z > 0