Question #122825

What do you understand by unit tangent vector

Expert's answer

Given a smooth vector-valued function "\\overrightarrow{r}(t)". Any vector parallel to "\\overrightarrow{r}'(t_0)" is *tangent* to the graph of "\\overrightarrow{r}(t)" at "t=t_0." It is often useful to consider just the *direction* of "\\overrightarrow{r}'(t)" and not its magnitude.

Therefore we are interested in the unit vector in the direction of "\\overrightarrow{r}'(t)"

This leads to a definition.

Let "\\overrightarrow{r}(t)" be a smooth function on an open interval "I." The unit tangent vector "\\overrightarrow{T}(t)" is

Let "\\overrightarrow{v}(t)=\\overrightarrow{r}'(t)" denote the velocity vector. Then we define the unit tangent vector* *by as the unit vector in the direction of the velocity vector.

The tangential component of acceleration is in the direction of the unit tangent vector

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