Does the vector function F––(t)=ti–+2tcostj–+2tsintk––𝐹_(𝑡)=𝑡𝑖_+2𝑡𝑐𝑜𝑠𝑡𝑗_+2𝑡𝑠𝑖𝑛𝑡𝑘_ lie on a paraboloid?
Question
Does the vector function
F––(t)=ti–+2tcostj–+2tsintk––𝐹_(𝑡)=𝑡𝑖_+2𝑡𝑐𝑜𝑠𝑡𝑗_+2𝑡𝑠𝑖𝑛𝑡𝑘_ lie on a paraboloid?
Solution
The vector function F(t) = ti + 2tcostj + 2tsintk does not lie on a paraboloid.
A paraboloid is a type of surface in three-dimensional Euclidean space. It is described by a quadratic equation in three variables. The general form of a paraboloid is given by the equation z = x^2 + y^2.
The vector function F(t) = ti + 2tcostj + 2tsintk is a parametric representation of a curve in three-dimensional space, not a surface. The x, y, and z coordinates of the curve are given by the functions x(t) = t, y(t) = 2tcost, and z(t) = 2tsint, respectively.
These functions do not satisfy the equation of a paraboloid. Therefore, the curve represented by the vector function F(t) does not lie on a paraboloid.
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