Vector Calculus Answers

An airplane flies at 400 miles per hour due east in a wind blowing at 50 miles per hour
from southeast. What is the flying speed and direction of the plane relative to the ground?

Given vector u = (-1, 1, 2) and vector v = (0, 1, 0).
a) u x v , and
b) Show that u x v is orthogonal to both u and v.

Given vector u = (0, 3, 3) and vector v = (-1, 1, 1).
a) Find the projection of u onto v, and
b) Find the vector component of u orthogonal to v.
[Give your answers in the form of linear combination of the standard unit vectors (i , j , and k ) ]

if a and b are perpendicular then prove that [a2+b2]=[a]2+[b]2

if a+b+c=0, |a|=3, |b|=5 and |c|=7, then find the angle between a and b.

Find the cosines of the angles that the line r = [-3,2,5] + t[2,2√2] makes with the coordinate axes

if magnitiude of three vectors ,what will be the direction between them so that the resultant will be zero?

Find the curl of the vector field A=ω ρ e^-( ρ/α )^2 uΦ ,where α and ω are constant . Both analytically and by use of the curl function .

Find the angles which the vector 3i-6j+3k makes with the coordinate axes.

Verify Stoke’s theorem for the vector
[b]A[/b]=(2x-y)[b]i[/b]-yz[sup]2[/sup][b]j[/b]-y[sup]2[/sup]z[b]k[/b],
& where S is the upper half surface of the sphere x[sup]2[/sup] + y[sup]2[/sup] + z[sup]2[/sup] = 1 and C is its boundary