For the following equation; r = ____ sin (t) i + ____ j + ____ cos (t) k, fill i

Question

For the following equation; r = ____ sin (t) i + ____ j + ____ cos (t) k, fill in the blanks to make the following space curves or surface. (If relevant, state the domain of t that you use]: a. Cylindrical helix b. Ellipse not residing entirely in the xy-, yz-, nor xz-planes c. Compressed cylindrical helix (i.e. resembling a Slinky at equilibrium) d. Paraboloidal helix (that is, a helix that lies on a paraboloid) e. Ellipsoid or hemiellipsoid (state which you chose) using variable s as well as t f A line segment two units long

Explanation / Answer

a> FOr a cylinderical helix

x = rsin(t) , y = ct and z = rcos(t) , for t E [0 , 2pi]

here r is the radius of the helix and 2*pi*c is the vertical separation between two loops.

lets take the radius as r = 1 and 2*pi*c = 2*pi , => c = 1

hence the equation becomes :

r = 1*sin(t) i + 1*tj + 1*cos(t)k

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