At Westonci.ca, we provide clear, reliable answers to all your questions. Join our vibrant community and get the solutions you need. Discover reliable solutions to your questions from a wide network of experts on our comprehensive Q&A platform. Get quick and reliable solutions to your questions from a community of experienced experts on our platform.
Sagot :
The parametric equation for the tangent line to the curve is x = 1 - t, y = t, z = 1 - t.
For this question,
The curve is given as
x(t)=e^-t cos(t),
y(t) =e^-t sin(t),
z(t)=e^-t
The point is at (1,0,1)
The vector equation for the curve is
r(t) = { x(t), y(t), z(t) }
Differentiate r(t) with respect to t,
x'(t) = -e^-t cos(t) + e^-t (-sin(t))
⇒ x'(t) = -e^-t cos(t) - e^-t sin(t)
⇒ x'(t) = -e^-t (cos(t) + sin(t))
y'(t) = - e^-t sin(t) + e^-t cos(t)
⇒ y'(t) = e^-t ((cos(t) - sin(t))
z'(t) = -e^-t
Then, r'(t) = { -e^-t (cos(t) + sin(t)), e^-t ((cos(t) - sin(t)), -e^-t }
The parameter value corresponding to (1,0,1) is t = 0. Putting in t=0 into r'(t) to solve for r'(t), we get
⇒ r'(t) = { -e^-0 (cos(0) + sin(0)), e^-0 ((cos(0) - sin(0)), -e^-0 }
⇒ r'(t) = { -1(1+0), 1(1-0), -1 }
⇒ r'(t) = { -1, 1, -1 }
The parametric equation for line through the point (x₀, y₀, z₀) and parallel to the direction vector <a, b, c > are
x = x₀+at
y = y₀+bt
z = z₀+ct
Now substituting the (x₀, y₀, z₀) as (1,0,1) and <a, b, c > into x, y and z, respectively to solve for the parametric equation of the tangent line to the curve, we get
x = 1 + (-1)t
⇒ x = 1 - t
y = 0 + (1)t
⇒ y = t
z = 1 + (-1)t
⇒ z = 1 - t
Hence we can conclude that the parametric equation for the tangent line to the curve is x = 1 - t, y = t, z = 1 - t.
Learn more about parametric equation here
https://brainly.com/question/24097871
#SPJ4
We hope this information was helpful. Feel free to return anytime for more answers to your questions and concerns. We appreciate your time. Please revisit us for more reliable answers to any questions you may have. We're glad you chose Westonci.ca. Revisit us for updated answers from our knowledgeable team.