Vector Calculus Answers

Prove that (a ⃗/a^2 - b ⃗/b^2 )^2= ((a ⃗ - b ⃗)/ab )^2

Find the vector of magnitude 3√2 which lies in ZX plane and is at right angle to the vector 2i ⃗+j ⃗+2k ⃗ .

If a ⃗=2i ⃗- j ⃗+ k ⃗ ,b ⃗= i ⃗- 3j ⃗-5k ⃗ find a vector c ⃗ such that a ⃗,b ⃗,c ⃗ form the sides of a right angled triangle taken in order .

magnitude of 3i-4j

evaluate ∇ × r(vector) / r(magnitude)

Find the direction in which the function f= x^2-y^2+2xy decreases more rapidly at the point (1,1)

Explain why it is not possible for two of a vector's direction angles to be less than 45°

if the force field defined by vector F=(3x^2yz-3y)i^ + (x^3z - 3x)j^ + (x^3y+2z)k^ conservative? if so, find the scalar potenial associated with the vector F.

A plane sheet of material is bound by the curve y = x^2 from x = 0 to x = 1 the x-axis
and the line x = 1. If the mass per unit area (density) of the sheet is xy find the mass of
the sheet

Show that
(vr .∇) vr = 1/2(∇v^2)− vr × (∇×vr)
Note: here vr= vector r