Easy Vectors and Hard Partial Differentials?

Question

1) ) Determine whether the vectors u = <-1, -2, 1>, v = <3, 0, -2> and a = <5, -4, 0> lie in the same plane when positioned so that their initial points coincide.

2a) Assume that f is a differentiable function of u, v and w. Let u=x - y, v=y - z and w=z - x. Verify that (∂f/∂x) plus (∂f/∂y) plus (∂f/∂z) = 0.

b) Suppose that z is defined implicitly as a function z = f(x,y) by the equation

x y z = cos(x plus y plus z). Find (∂z/∂x) and also (∂z/∂y).

Answer 1

u x v = 〈 4 , 1 , 6 〉

u x a = 〈 4 , 5 , 14 〉

since they have different cross-products, getting the lines two at a time forms a different plane each time.

.. . .

2.

∂f/∂x = ∂f/∂u * (1) + ∂f/∂v * (0) + ∂f/∂w * (-1)

get the rest of the partials and see what happens when you combine them