Analyzing the Helix r(t) = <cos t, sin t, t>

Calculus 3 · Axiom Academy

Differentiate the position vector, find its constant speed, build the unit tangent, then compute the curvature. The components and trace a radius-1 circle in the x - y plane while z = t carries the point steadily upward — a spring-like curve. The start point at t = 0 is (1, 0, 0) , and every full turn raises the height by . Analyze the circular helix : find the velocity , its speed , the unit tangent vector , and the curvature . Nice work — you fully analyzed the circular helix from velocity through curvature. Here's what carried the solution: Velocity vector: differentiate each component of to get . Constant speed: — the t dropped out, so the helix is traced at constant speed. Curvature: — constant, since a helix curves the same at every point. Pythagorean identity: is the workhorse that collapses the trig terms in both the speed and the cross product. The same toolkit — velocity, speed, unit tangent, curvature — describes any smooth space curve. The helix is special because all three of these quantities turn out to be constant.

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