Give the curve of the following vector equation.?

Question

I understand that the curve is a spiral, and that x^2+y^2=1. But what happens to x=t? if that becomes one, how?

Answer 1

from y^2 + z^2 = cos^2t + sin^2t = 1, the circular part is with respect to y and z. So, when t = 0, you're at (0, 1, 0) and as t increases, x increases as t and the curve follows a spiriling path with the x axis as the axis of the spiral.

Answer 2

Are you sure the vector is not
where x = cos(2t) y = sin 2t and t = z?

If it is then this is the equation of a helix. If t were a constant, it would be a circle at a distance of the constant above or below th XY plane. But since t is a variable, the curve becomes a helix and looks like a spring stretcked out vertically.