Solution r(t) = 6 cos(t) i + 6 sin(t)j + 3tk. The parametric equations for this curve are x= y = 6 sin(t), z = Since x² + y² = + 36.sin²(t) = the curve must lie on the circular cylinder x² + y² = The point (x, y, z) lies directly above the point (x, y, 0), which moves counterclockwise around the circle x² + y2. in the xy-plane. (The projection of the curve onto the xy-plane has vector equation r(t) = (6 cos(t), 6 sin(t), 0). See this example.) Since z = 3t, the curve spirals upward around the cylinder as t increases. The curve, shown in the figure below, is called a helix.
Solution r(t) = 6 cos(t) i + 6 sin(t)j + 3tk. The parametric equations for this curve are x= y = 6 sin(t), z = Since x² + y² = + 36.sin²(t) = the curve must lie on the circular cylinder x² + y² = The point (x, y, z) lies directly above the point (x, y, 0), which moves counterclockwise around the circle x² + y2. in the xy-plane. (The projection of the curve onto the xy-plane has vector equation r(t) = (6 cos(t), 6 sin(t), 0). See this example.) Since z = 3t, the curve spirals upward around the cylinder as t increases. The curve, shown in the figure below, is called a helix.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:Sketch the curve whose vector equation is
Solution
r(t) = 6 cos(t) i + 6 sin(t) j + 3tk.
The parametric equations for this curve are
X =
I
y = 6 sin(t), z =
Since x² + y² =
+ 36. sin²(t) =
The point (x, y, z) lies directly above the point (x, y, 0), which
moves counterclockwise around the circle x² + y2 =
in the xy-plane. (The projection of the curve onto the xy-plane has vector equation r(t) = (6 cos(t), 6 sin(t), 0). See this
example.) Since z = 3t, the curve spirals upward around the cylinder as t increases. The curve, shown in the figure below, is called a helix.
ZA
(6, 0, 0)
(0, 6, 37)
I
the curve must lie on the circular cylinder x² + y² =
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