# Why does $\vec{r}\,'(t)$ give the tangent vector to $\vec{r}(t)$ if $\vec{r}\,'(t)$ is orthogonal to $\vec{r}(t)$

I have seen a proof that $$\vec{r}\,'(t)$$ is orthogonal to $$\vec{r}(t)$$, but $$\vec{r}\,'(t)$$ gives the tangent vector to the curve $$\vec{r}(t)$$, for any $$t$$. I don't understand how $$\vec{r}\,'(t)$$ can represent the tangent vector but also be orthogonal to the vector curve?

Just another question related to the above: If the binormal vector is defined to be $$\vec{B}(t) = \vec{T}(t) \times \vec{N}(t),$$ where $$\vec{T}(t)$$ and $$\vec{N}(t)$$ represent the unit tangent and unit normal vector respectively, does the binormal vector give a unit vector orthogonal to both the tangent and normal vector, in the direction according to the 'right hand rule'? What significance does this have?

• In general the derivative of the position vector is not orthogonal to the position vector. Are you thinking of the derivative of the unit vector being orthogonal to the position vector? Commented Mar 7, 2019 at 22:15
• Consider a circle: in such a case, the tangent vector is orthogonal to the position vector. Commented Mar 7, 2019 at 22:19
• @Andrew I was just on this page, where it gives a fact about the orthogonality about the derivative of a position vector, but I don't think I understand it. Commented Mar 7, 2019 at 22:25
• That is assuming that $\left\|\vec{r}(t)\right\|$ is constant over time. In general though, $\vec{r}(t)$ need not be orthogonal to $\vec{r}'(t)$. However, it is certainly possible to happen, for example if $\vec{r}(t)$ is tracing out a circle centred at the origin. Commented Mar 7, 2019 at 22:27
• @MinusOne-Twelfth , like a circle, centred at the origin? Commented Mar 7, 2019 at 22:27

## 1 Answer

To help make sense of this think of $$r(t)$$ as your position as you are driving around on earth (approximated as a sphere). The tangent vector $$T(t)$$ is the direction your car is facing. The normal vector $$N(t)$$ is the direction you're turning your steering wheel (left or right). The binormal vector is a vector orthogonal to both (pointing up or down).