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ParametricPlot3D

3D parametric plot.

$ wo 'Head[ParametricPlot3D[{Cos[t], Sin[t], t}, {t, 0, 6}]]'
Graphics3D

A list of triples draws one curve per triple. The items only have to take triple shape once evaluated, so a solution can be substituted into each of them — how a trajectory is drawn beside its two coordinate projections:

$ wo "s = NDSolve[{x'[t] == -x[t], y'[t] == x[t] - y[t], x[0] == 1, y[0] == 0}, {x[t], y[t]}, {t, 0, 3}]; Head[ParametricPlot3D[{{0, x[t], y[t]} /. s, {t, x[t], 0} /. s, {t, 2, y[t]} /. s}, {t, 0, 3}]]"
Graphics3D

PlotStyle -> Tube[r] draws the curve as a tube of radius r rather than as a line; any colour given alongside it still applies:

$ wo 'Head[ParametricPlot3D[{Cos[t], Sin[t], t/5}, {t, 0, 6.2}, PlotStyle -> {Red, Tube[0.2]}]]'
Graphics3D