Part 5/6:
Understanding the unit tangent vector is crucial because it gives the direction of the curve at a particular point, independent of the speed at which the curve is traversed. It is fundamental in analyzing motion along a space curve, such as in physics for velocity directions, or in differential geometry for understanding the curvature and torsion of the path.
Approximate Position on the Curve at ( t=0 )
Alongside the tangent vector, it's often helpful to note the point on the curve corresponding to ( t=0 ):
[
r(0) = (1 + 0^3) \mathbf{i} + 0 \times e^{0} \mathbf{j} + \sin(0) \mathbf{k} = 1 \mathbf{i} + 0 \mathbf{j} + 0 \mathbf{k} = (1, 0, 0)
]
Thus, the point on the curve at ( t=0 ) is ( (1, 0, 0) ).