ParametricPlot3D[Cos[t](1/(Sqrt[3])){1,1,1}+Sin[t](1/(Sqrt[2])){1,-1,0},{t,0,2 Pi}]

To add vectors we need to use Graphics3D. I will also name the vectors.

vu=(1/(Sqrt[2])){1,-1,0}(* red, maroon *); vv= (1/(Sqrt[3])){1,1,1}(* green *);  Show[Graphics3D[{
{Glow[RGBColor[0.5,0,0]],RGBColor[0.5,0,0],Arrowheads[0.04],FaceForm[RGBColor[0.25,0,0]],Arrow[Tube[{{0,0,0},vu},0.015]]},
{Glow[RGBColor[0,0.5,0]],RGBColor[0,0.5,0],Arrowheads[0.04],FaceForm[RGBColor[0,0.25,0]],Arrow[Tube[{{0,0,0},vv},0.015]]}
}],ParametricPlot3D[Cos[t]vu+Sin[t]vv,{t,0,2 Pi},PlotStyle->{{Thickness[0.007],RGBColor[0.25,0.25,.25]}}],PlotRange->{{-1,1},{-1,1},{-1,1}},Axes->True]

It is nice to place one more more vector in this plot:

vu=(1/(Sqrt[2])){1,-1,0}(* red, maroon *); vv= (1/(Sqrt[3])){1,1,1}(* green *); vw= (1/(Sqrt[6])){-1,-1,2}(* blue, navy *);  
 Show[Graphics3D[{
{Glow[RGBColor[0.5,0,0]],RGBColor[0.5,0,0],Arrowheads[0.04],FaceForm[RGBColor[0.25,0,0]],Arrow[Tube[{{0,0,0},vu},0.015]]},
{Glow[RGBColor[0,0.5,0]],RGBColor[0,0.5,0],Arrowheads[0.04],FaceForm[RGBColor[0,0.25,0]],Arrow[Tube[{{0,0,0},vv},0.015]]},
{Glow[RGBColor[0,0,0.5]],RGBColor[0,0,0.5],Arrowheads[0.04],FaceForm[RGBColor[0,0,0.5]],Arrow[Tube[{{0,0,0},vw},0.015]]}
}],
ParametricPlot3D[Cos[t]vu+Sin[t]vv,{t,0,2 Pi},PlotStyle->{{Thickness[0.007],RGBColor[0.25,0.25,.25]}}],PlotRange->{{-1,1},{-1,1},{-1,1}},Axes->True]

vu=(1/(Sqrt[2])){1,-1,0}(* red, maroon *); vv= (1/(Sqrt[3])){1,1,1}(* green *); vw= (1/(Sqrt[6])){-1,-1,2}(* blue, navy *);  
 Manipulate[
Show[
Graphics3D[{
{Glow[RGBColor[0.5,0,0]],RGBColor[0.5,0,0],Arrowheads[0.04],FaceForm[RGBColor[0.25,0,0]],Arrow[Tube[{{0,0,0},vu},0.015]]},
{Glow[RGBColor[0,0.5,0]],RGBColor[0,0.5,0],Arrowheads[0.04],FaceForm[RGBColor[0,0.25,0]],Arrow[Tube[{{0,0,0},vv},0.015]]},
{Glow[RGBColor[0,0,0.5]],RGBColor[0,0,0.5],Arrowheads[0.04],FaceForm[RGBColor[0,0,0.5]],Arrow[Tube[{{0,0,0},vw},0.015]]}
}],
ParametricPlot3D[Cos[s]vu+Sin[s]vv,{s,0,2 Pi},PlotStyle->{{Thickness[0.001],RGBColor[0.25,0.25,.25]}}],
Graphics3D[{{PointSize[0.02],Point[Cos[t]vu+Sin[t]vv]}}],
PlotRange->{{-1,1},{-1,1},{-1,1}},Axes->True],{t,0,2 Pi}]

Now we could utilize the blue vector by making a helix

vu=(1/(Sqrt[2])){1,-1,0}(* red, maroon *); vv= (1/(Sqrt[3])){1,1,1}(* green *); vw= (1/(Sqrt[6])){-1,-1,2}(* blue, navy *);  
 Show[Graphics3D[{
{Glow[RGBColor[0.5,0,0]],RGBColor[0.5,0,0],Arrowheads[0.04],FaceForm[RGBColor[0.25,0,0]],Arrow[Tube[{{0,0,0},vu},0.015]]},
{Glow[RGBColor[0,0.5,0]],RGBColor[0,0.5,0],Arrowheads[0.04],FaceForm[RGBColor[0,0.25,0]],Arrow[Tube[{{0,0,0},vv},0.015]]},
{Glow[RGBColor[0,0,0.5]],RGBColor[0,0,0.5],Arrowheads[0.04],FaceForm[RGBColor[0,0,0.5]],Arrow[Tube[{{0,0,0},vw},0.015]]}
}],
ParametricPlot3D[Cos[t]vu+Sin[t]vv+(t/(4 Pi)) vw,{t,0,4 Pi},PlotStyle->{{Thickness[0.007],RGBColor[0.25,0.25,.25]}}],PlotRange->{{-2,2},{-2,2},{-2,2}},Axes->True]

vu=(1/(Sqrt[2])){1,-1,0}(* red, maroon *); vv= (1/(Sqrt[3])){1,1,1}(* green *); vw= (1/(Sqrt[6])){-1,-1,2}(* blue, navy *);  
 Manipulate[
Show[
Graphics3D[{
{Glow[RGBColor[0.5,0,0]],RGBColor[0.5,0,0],Arrowheads[0.04],FaceForm[RGBColor[0.25,0,0]],Arrow[Tube[{{0,0,0},vu},0.015]]},
{Glow[RGBColor[0,0.5,0]],RGBColor[0,0.5,0],Arrowheads[0.04],FaceForm[RGBColor[0,0.25,0]],Arrow[Tube[{{0,0,0},vv},0.015]]},
{Glow[RGBColor[0,0,0.5]],RGBColor[0,0,0.5],Arrowheads[0.04],FaceForm[RGBColor[0,0,0.5]],Arrow[Tube[{{0,0,0},vw},0.015]]}
}],
ParametricPlot3D[Cos[s]vu+Sin[s]vv+(s/(4 Pi)) vw,{s,0,4 Pi},PlotStyle->{{Thickness[0.001],RGBColor[0.25,0.25,.25]}}],
Graphics3D[{{PointSize[0.02],Point[Cos[t]vu+Sin[t]vv+(t/(4 Pi)) vw]}}],
PlotRange->{{-2,2},{-2,2},{-2,2}},Axes->True],{t,0,4 Pi}]

This is a graph of the sine function placed in 3-dimensional space.

ParametricPlot3D[t(1/(Sqrt[6])){-1,-1,2}+Sin[t](1/(Sqrt[3])){1,1,1},{t,-2 Pi,2 Pi}]

Here we emphasise the vectors used in the previous plot.

vu=(1/(Sqrt[2])){1,-1,0}(* red, maroon *); vv= (1/(Sqrt[3])){1,1,1}(* green *); vw= (1/(Sqrt[6])){-1,-1,2}(* blue, navy *);  
 Show[Graphics3D[{
{Glow[RGBColor[0.5,0,0]],RGBColor[0.5,0,0],Arrowheads[0.04],FaceForm[RGBColor[0.25,0,0]],Arrow[Tube[{{0,0,0},vu},0.015]]},
{Glow[RGBColor[0,0.5,0]],RGBColor[0,0.5,0],Arrowheads[0.04],FaceForm[RGBColor[0,0.25,0]],Arrow[Tube[{{0,0,0},vv},0.015]]},
{Glow[RGBColor[0,0,0.5]],RGBColor[0,0,0.5],Arrowheads[0.04],FaceForm[RGBColor[0,0,0.5]],Arrow[Tube[{{0,0,0},vw},0.015]]}
}],
ParametricPlot3D[t vw+Sin[t]vv,{t,-2 Pi,2 Pi},PlotStyle->{{Thickness[0.007],RGBColor[0.25,0.25,.25]}}],PlotRange->{{-3,3},{-3,3},{-4,4}},Axes->True]

Replace the green vector with the red one.

vu=(1/(Sqrt[2])){1,-1,0}(* red, maroon *); vv= (1/(Sqrt[3])){1,1,1}(* green *); vw= (1/(Sqrt[6])){-1,-1,2}(* blue, navy *);  
 Show[Graphics3D[{
{Glow[RGBColor[0.5,0,0]],RGBColor[0.5,0,0],Arrowheads[0.04],FaceForm[RGBColor[0.25,0,0]],Arrow[Tube[{{0,0,0},vu},0.015]]},
{Glow[RGBColor[0,0.5,0]],RGBColor[0,0.5,0],Arrowheads[0.04],FaceForm[RGBColor[0,0.25,0]],Arrow[Tube[{{0,0,0},vv},0.015]]},
{Glow[RGBColor[0,0,0.5]],RGBColor[0,0,0.5],Arrowheads[0.04],FaceForm[RGBColor[0,0,0.5]],Arrow[Tube[{{0,0,0},vw},0.015]]}
}],
ParametricPlot3D[t vw+Sin[t]vu,{t,-2 Pi,2 Pi},PlotStyle->{{Thickness[0.007],RGBColor[0.25,0.25,.25]}}],PlotRange->{{-3,3},{-3,3},{-4,4}},Axes->True]

Now replace the red vector with a linear combination of red and green.

s=1;vu=(1/(Sqrt[2])){1,-1,0}(* red, maroon *); vv= (1/(Sqrt[3])){1,1,1}(* green *); vw= (1/(Sqrt[6])){-1,-1,2}(* blue, navy *);  
 Show[Graphics3D[{
{Glow[RGBColor[0.5,0,0]],RGBColor[0.5,0,0],Arrowheads[0.04],FaceForm[RGBColor[0.25,0,0]],Arrow[Tube[{{0,0,0},vu},0.015]]},
{Glow[RGBColor[0,0.5,0]],RGBColor[0,0.5,0],Arrowheads[0.04],FaceForm[RGBColor[0,0.25,0]],Arrow[Tube[{{0,0,0},vv},0.015]]},
{Glow[RGBColor[0,0,0.5]],RGBColor[0,0,0.5],Arrowheads[0.04],FaceForm[RGBColor[0,0,0.5]],Arrow[Tube[{{0,0,0},vw},0.015]]},
{Glow[RGBColor[Abs[Cos[s]],Abs[Sin[s]],0]],RGBColor[Abs[Cos[s]],Abs[Sin[s]],0],Arrowheads[0.04],FaceForm[RGBColor[Abs[Cos[s]],Abs[Sin[s]],0]],Arrow[Tube[{{0,0,0},(Cos[s] vu+Sin[s] vv)},0.015]]}
}],
ParametricPlot3D[t vw+Sin[t](Cos[s] vu+Sin[s] vv),{t,-2 Pi,2 Pi},PlotStyle->{{Thickness[0.007],RGBColor[0.25,0.25,.25]}}],PlotRange->{{-3,3},{-3,3},{-4,4}},Axes->True]

Now manipulate the previous plot

vu=(1/(Sqrt[2])){1,-1,0}(* red, maroon *); vv= (1/(Sqrt[3])){1,1,1}(* green *); vw= (1/(Sqrt[6])){-1,-1,2}(* blue, navy *);  
 Manipulate[Show[Graphics3D[{
{Glow[RGBColor[0.5,0,0]],RGBColor[0.5,0,0],Arrowheads[0.04],FaceForm[RGBColor[0.25,0,0]],Arrow[Tube[{{0,0,0},vu},0.015]]},
{Glow[RGBColor[0,0.5,0]],RGBColor[0,0.5,0],Arrowheads[0.04],FaceForm[RGBColor[0,0.25,0]],Arrow[Tube[{{0,0,0},vv},0.015]]},
{Glow[RGBColor[0,0,0.5]],RGBColor[0,0,0.5],Arrowheads[0.04],FaceForm[RGBColor[0,0,0.5]],Arrow[Tube[{{0,0,0},vw},0.015]]},
{Glow[RGBColor[Abs[Cos[s]],Abs[Sin[s]],0]],RGBColor[Abs[Cos[s]],Abs[Sin[s]],0],Arrowheads[0.04],FaceForm[RGBColor[Abs[Cos[s]],Abs[Sin[s]],0]],Arrow[Tube[{{0,0,0},(Cos[s] vu+Sin[s] vv)},0.015]]}
}],
ParametricPlot3D[t vw+Sin[t](Cos[s] vu+Sin[s] vv),{t,-2 Pi,2 Pi},PlotStyle->{{Thickness[0.007],RGBColor[0.25,0.25,.25]}}],PlotRange->{{-3,3},{-3,3},{-4,4}},Axes->True],{s,0,2Pi}]

Now replace the Sin[t] function with  (2+Sin[t])

vu=(1/(Sqrt[2])){1,-1,0}(* red, maroon *); vv= (1/(Sqrt[3])){1,1,1}(* green *); vw= (1/(Sqrt[6])){-1,-1,2}(* blue, navy *);  
 Manipulate[Show[Graphics3D[{
{Glow[RGBColor[0.5,0,0]],RGBColor[0.5,0,0],Arrowheads[0.02],FaceForm[RGBColor[0.25,0,0]],Arrow[Tube[{{0,0,0},vu},0.015]]},
{Glow[RGBColor[0,0.5,0]],RGBColor[0,0.5,0],Arrowheads[0.02],FaceForm[RGBColor[0,0.25,0]],Arrow[Tube[{{0,0,0},vv},0.015]]},
{Glow[RGBColor[0,0,0.5]],RGBColor[0,0,0.5],Arrowheads[0.02],FaceForm[RGBColor[0,0,0.5]],Arrow[Tube[{{0,0,0},vw},0.015]]},
{Glow[RGBColor[Abs[Cos[s]],Abs[Sin[s]],0]],RGBColor[Abs[Cos[s]],Abs[Sin[s]],0],Arrowheads[0.02],FaceForm[RGBColor[Abs[Cos[s]],Abs[Sin[s]],0]],Arrow[Tube[{{0,0,0},(Cos[s] vu+Sin[s] vv)},0.015]]}
}],
ParametricPlot3D[t vw+(2+Sin[t])(Cos[s] vu+Sin[s] vv),{t,-2 Pi,2 Pi},PlotStyle->{{Thickness[0.007],RGBColor[0.25,0.25,.25]}}],PlotRange->{{-5,5},{-5,5},{-6,6}},Axes->True],{s,0,2Pi}]

Now I will replace the vectors vu, vv, and vw with the coordinate vectors  vi, vj, vk

vi={1,0,0}(* red, maroon *); vj= {0,1,0}(* green *); vk={0,0,1}(* blue, navy *);  
 Manipulate[Show[Graphics3D[{
{Glow[RGBColor[0.5,0,0]],RGBColor[0.5,0,0],Arrowheads[0.02],FaceForm[RGBColor[0.25,0,0]],Arrow[Tube[{{0,0,0},vi},0.015]]},
{Glow[RGBColor[0,0.5,0]],RGBColor[0,0.5,0],Arrowheads[0.02],FaceForm[RGBColor[0,0.25,0]],Arrow[Tube[{{0,0,0},vj},0.015]]},
{Glow[RGBColor[0,0,0.5]],RGBColor[0,0,0.5],Arrowheads[0.02],FaceForm[RGBColor[0,0,0.5]],Arrow[Tube[{{0,0,0},vk},0.015]]},
{Glow[RGBColor[Abs[Cos[s]],Abs[Sin[s]],0]],RGBColor[Abs[Cos[s]],Abs[Sin[s]],0],Arrowheads[0.02],FaceForm[RGBColor[Abs[Cos[s]],Abs[Sin[s]],0]],Arrow[Tube[{{0,0,0},(Cos[s] vi+Sin[s] vj)},0.015]]}
}],
ParametricPlot3D[t vk+(2+Sin[t])(Cos[s] vi+Sin[s] vj),{t,-2 Pi,2 Pi},PlotStyle->{{Thickness[0.007],RGBColor[0.25,0.25,.25]}}],PlotRange->{{-4,4},{-4,4},{-6,6}},Axes->True],{s,0,2Pi}]

Now I will place a helix around the z-axis and than rotate it.

vi={1,0,0}(* red, maroon *); vj= {0,1,0}(* green *); vk={0,0,1}(* blue, navy *);  
 Show[Graphics3D[{
{Glow[RGBColor[0.5,0,0]],RGBColor[0.5,0,0],Arrowheads[0.02],FaceForm[RGBColor[0.25,0,0]],Arrow[Tube[{{0,0,0},vi},0.015]]},
{Glow[RGBColor[0,0.5,0]],RGBColor[0,0.5,0],Arrowheads[0.02],FaceForm[RGBColor[0,0.25,0]],Arrow[Tube[{{0,0,0},vj},0.015]]},
{Glow[RGBColor[0,0,0.5]],RGBColor[0,0,0.5],Arrowheads[0.02],FaceForm[RGBColor[0,0,0.5]],Arrow[Tube[{{0,0,0},vk},0.015]]}
}],
ParametricPlot3D[(t/Pi) vk+(Cos[t])(vi)+(Sin[t])(vj),{t,-3 Pi,3 Pi},PlotStyle->{{Thickness[0.007],RGBColor[0.25,0.25,.25]}}],PlotRange->{{-2,2},{-2,2},{-3,3}},Axes->True]

Now rotate the above helix:

vi={1,0,0}(* red, maroon *); vj= {0,1,0}(* green *); vk={0,0,1}(* blue, navy *);  
 Manipulate[Show[Graphics3D[{
{Glow[RGBColor[0.5,0,0]],RGBColor[0.5,0,0],Arrowheads[0.02],FaceForm[RGBColor[0.25,0,0]],Arrow[Tube[{{0,0,0},vi},0.015]]},
{Glow[RGBColor[0,0.5,0]],RGBColor[0,0.5,0],Arrowheads[0.02],FaceForm[RGBColor[0,0.25,0]],Arrow[Tube[{{0,0,0},vj},0.015]]},
{Glow[RGBColor[0,0,0.5]],RGBColor[0,0,0.5],Arrowheads[0.02],FaceForm[RGBColor[0,0,0.5]],Arrow[Tube[{{0,0,0},vk},0.015]]},
{Glow[RGBColor[Abs[Cos[s]],Abs[Sin[s]],0]],RGBColor[Abs[Cos[s]],Abs[Sin[s]],0],Arrowheads[0.02],FaceForm[RGBColor[Abs[Cos[s]],Abs[Sin[s]],0]],Arrow[Tube[{{0,0,0},(Cos[s] vi+Sin[s] vj)},0.015]]},
{Glow[RGBColor[Abs[Sin[s]],Abs[Cos[s]],0]],RGBColor[Abs[Sin[s]],Abs[Cos[s]],0],Arrowheads[0.02],FaceForm[RGBColor[Abs[Sin[s]],Abs[Cos[s]],0]],Arrow[Tube[{{0,0,0},(-Sin[s] vi+Cos[s] vj)},0.015]]}
}],
ParametricPlot3D[(t/Pi) vk+(Cos[t])(Cos[s] vi+Sin[s] vj)+(Sin[t])(-Sin[s] vi+Cos[s] vj),{t,-3 Pi,3 Pi},PlotStyle->{{Thickness[0.007],RGBColor[0.25,0.25,.25]}}],PlotRange->{{-2,2},{-2,2},{-3,3}},Axes->True],{s,0,2Pi}]