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You can introduce a new parameter s that will parametrize the line. With p=(-e^pi,0,e^pi), the line is then p+s*tvec(t=t0) for some fixed value t0 of the parameter t.

Then, the line can be drawn as follows.

var('s,t')
p=vector([-exp(pi),0,exp(pi)])
tvec = vector((e^t * cos(t) - e^t * sin(t), e^t * sin(t) + e^t * cos(t), e^t))
parametric_plot3d(p+s*tvec(t=pi/2),(s,0,5),thickness=3)