How do you know if the tangent unit vector parallel

Question

Hi I have been given the following questions and answers (I've only included the part relevant for my question) on work done. My question is actually to do with the solutions. Here the unit tangent vector in Q1 is perpendicular to F and in Q2 parallel to F. My question is how do we know that? Any help would be appreciated:) Thank you

Questions:

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Q1 Let us compute the work done by the force field $\overrightarrow F = x \hat i+y\hat j$ along a circle $C_a$ of radius a > 0 centred at the origin (transversed counterclockwise).

If $\hat T$ denotes the unit vector tangent to the circle Ca, then $\overrightarrow F$ ⊥ $\hat T$ and so $\overrightarrow F$·$\hat T=0$

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Q2 Let us now compute the work done by the force field $\overrightarrow F = -y \hat i+x\hat j$ along the same circle $C_a$ of radius a > 0 centred at the origin (transversed counterclockwise).

In this case, $ \overrightarrow F \parallel \hat T$

Answer 1

One has to know what the tangent vector to a circle centered at the origin is. Once one knows that, checking that out is orthogonal to the vector field in question 1, and parallel to the vector field in question 2 are rather simple tasks.

The tangent vector (in the counterclockwise direction) at the point $(x,y)$ to a circle centered at the origin is $-yi+xj$. This can be found, for instance, by parameterising the circle using your favourite parametrisation, differentiating this parametrisation, and inserting the relevant value of the parameter. Or one can note that the tangent vector is necessarily orthogonal to the radius, and the radius is $xi+yj$.

Answer 2

Set $x=a\cos t$ and $y=a\sin t.$ This describes the circle. Then find the unit tangent to this circle. To verify perpendicularity or parallelism, take dot and cross products respectively, and see that they vanish.