Question

Does the magnitude of a vector equal the square root of the dot product of its components?

Answer 1

dot product of the components is always 0

so if a and b are the components of a vector
dot product (a.b) = |a||b|cos (theta)
so theta is 90 degree or pi/2 for each case and hence dot product is zero (why ?)

hence the answer is NO

Answer 2

That's a pretty sexy proof