Mathematics

Mathematics

Problem

Find parametric equations for the tangent line to the curve with the given parametric equations at the specified point. x=e−tcos⁡t,y=e−tsin⁡t,z=e−t;(1,0,1)x = e^{-t} \cos t, y = e^{-t} \sin t, z = e^{-t}; (1, 0, 1)
Intuitive explanation

The key is finding the direction vector of the tangent line by taking derivatives of each parametric equation. First, identify which tt-value produces the given point, then compute dxdt\frac{dx}{dt}, dydt\frac{dy}{dt}, and dzdt\frac{dz}{dt} at that moment—this velocity vector points along the curve. Finally, use the point and direction vector to write the tangent line's parametric form.

Intuitive explanation

The key is finding the direction vector of the tangent line by taking derivatives of each parametric equation. First, identify which tt-value produces the given point, then compute dxdt\frac{dx}{dt}, dydt\frac{dy}{dt}, and dzdt\frac{dz}{dt} at that moment—this velocity vector points along the curve. Finally, use the point and direction vector to write the tangent line's parametric form.

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