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DETAILS SCALCETOM 13.1.AE.004. MY NOTES ASK YOUR TEACHER EXAMPLE 4 Sketch the curve whose vector equation is r(t) = 4 cos(t) i + 4 sin(t) j + 2tk. SOLUTION The parametric equations for this curve are X y = 4 sin(t), z = Since x2 + y2 : + 16 sin (t) = , the curve must lie on the circular cylinder x2 + 12 = The point (x, y, z) lies directly above the point (x, y, 0), which moves counterclockwise around the circle x2 + y? = in the xy-plane. (The projection of the curve onto the xy-plane has vector equation r(t) = (4 cos(t), 4 sin(t), 0). See this example.) Since z = 2t, the curve spirals upward around the xX cylinder as t increases. The curve, shown in the figure, is called a helix. Video Example ()