Vector Analysis Problem No.2

Given the vector field F=0.4(y-2x)ax-[200/(x²+y²+z²)]az; (a) evaluate |F| at P(-4,3,5); (b) find a unit vector specifying the direction of F at P. Describe the locus of all points for which: (c) Fx=1; (d) |Fz|=2.

Solutions:

(a) evaluate |F| at P(-4,3,5):

Substitute coordinates of point P to the given equation.

F = (0.4y-0.7x)ax-[200/(x²+y²+z²)]az

F = [0.4(3)-0.8(-4)]ax-[-200/(4²+3²+5²)]az

F = (1.2+3.2)ax-(200/50)az

F = 4.4ax-4az

|F| = √[(4.4²)+4²] = 5.95

(b) find a unit vector specifying the direction of F at P:

a_F = F/|F| = (4.4ax-4az)/5.95

a_F = 0.740ax-0.673az

(c) Describe the locus of all points for which Fx = 1:

F = Fxax-Fzaz

Fx = 0.4(y-2x)

1 = 0.4y-0.8x

0.4y = 0.8x+1

y = (0.8x+1)/0.4

y = 2x+2.5 (equation of a plane) 

(d) Describe the locus of all points for which |Fz| = 2:

|Fz| = √Fz² = Fz

2 = 200/(x²+y²+z²) 

(x²+y²+z²) = 200/2

(x²+y²+z²) = 100 (equation of a sphere)