Given the parametric equation $r\left(t\right)=\left(\cos \right(t),\sin (t),t)$, where does the helix lie?

• A. On the surface of a sphere of radius $1$ centered at the origin
• B. On the surface of a cylinder of radius $1$ centered along the z-axis
• C. On the surface of a cone with vertex at the origin
• D. On the $xy$-plane

Answer: (B)

Verified step by step guidance

1. Identify the parametric equations given: \(r(t) = (\cos(t), \sin(t), t)\), where the components correspond to \(x = \cos(t)\), \(y = \sin(t)\), and \(z = t\).
2. Focus on the \(x\) and \(y\) components to understand the projection of the curve onto the \(xy\)-plane. Notice that \(x = \cos(t)\) and \(y = \sin(t)\) satisfy the equation \(x^2 + y^2 = \cos^2(t) + \sin^2(t)\).
3. Recall the Pythagorean identity \(\cos^2(t) + \sin^2(t) = 1\), which means that for all values of \(t\), the point \((x, y)\) lies on the circle of radius 1 centered at the origin in the \(xy\)-plane.
4. Since \(z = t\) varies independently and can take any real value, the curve extends infinitely along the \(z\)-axis while maintaining a constant radius of 1 in the \(xy\)-plane.
5. Conclude that the helix lies on the surface of a cylinder of radius 1 centered along the \(z\)-axis, because the distance from the \(z\)-axis (given by \(\sqrt{x^2 + y^2}\)) is always 1.