Suppose r(t)= cos(t)i+sin(t)j+(2t)k represents the position of a particle on a helix, where Z is the height of the particle abov

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Suppose r(t)= cos(t)i+sin(t)j+(2t)k represents the position of a particle on a helix, where Z is the height of the particle abov

e the ground. When it is 20 units above the ground, the particle leaves the helix and moves along the tangent. Find parametric equations for this tangent line (pick t so that it is continuous through the time when the particle leaves the helix).

Mathematics

1 answer:

Inessa [12.5K] · Answer 1 · 2026-03-24T21:43:20+03:00

The parametric equations defining the tangent line are:x = Cos(10) - t×Sin(10)

y = Sin(10) + t×Cos(10)

z = 20 + 2t

Step-by-step explanation:

When Z=20:Z=2t=20 ⇒ t=10

The point where tangency occurs is:

r(10)= Cos(10)

i

+ Sin(10)

j

+ 20 kTo derive the tangent line, we need the derivative of r(t):r'(t)= -Sin(t)

i

+ Cos(t)

j

+ 2 kThe directional vector at t=10 yields:r'(10)= -Sin(10) i

+ Cos(10)

j

+ 2

kThus, the equation for the tangent line can be formatted as:x = cos (10) - t×Sin(10)

y = sin (10) + t×Cos(10)

z = 20 + 2t

Suppose r(t)= cos(t)i+sin(t)j+(2t)k represents the position of a particle on a helix, where Z is the height of the particle abov

Explanation: