A space curve is defined by C: r(s) = 2si + s^3j + (5s^2 + 3s)k. Find the Cartes

Question

A space curve is defined by C: r(s) = 2si + s^3j + (5s^2 + 3s)k. Find the Cartesian form of the equation for the plane that is perpendicular to the space curve C at the point where s = 3 Your answer should be an equation, expressed in terms of the Cartesian variables x, y and z using the correct syntax. For example: 3*x-2*y + 5*z = 2, or, 2*(x-1) + 4*(y-2) + z-1 = 0, or 3*x + 6*z = 12-y, or y - x + 35*(z-256) = 20 Do not use decimal approximations all numbers should be entered as exact expressions, for example 5/2

Explanation / Answer

r (s) = 2s i + s^3 j + (5s^2 + 3s) k

tangent vector =

2 i + 3s^2 j + (10s+ 3) k

hence tangent at s = 3

(2,27,23)

at s = 3

r(s) = 6 i + 27 j + 54 k

passing through point (6,27,54)

tangent vector at the point: (2,27,23)

equation of plane be 2(x-6)+27(y-27) +23(z-54) = 0

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