For a regular parametrised plane curve $\alpha$, show that $\langle \alpha''(t),n(t)\rangle =- \langle \alpha'(t),n'(t)\rangle$

Question

When I was proving some properties of regular parametrised plane curve $\alpha:I\to R^2$ which has a normal vector $n(t)$, I encountered the need to prove the following:

$$\langle \alpha''(t),n(t)\rangle =- \langle \alpha'(t),n'(t)\rangle$$

which then troubled me.

Do I need to set the components of each vector and do the dot product by multiplying each component then sum them up? Or is there any identities or other better way to show $\langle \alpha''(t),n(t)\rangle =- \langle \alpha'(t),n'(t)\rangle$ ? Or is there any specific property of dot product that I can use?

Many thanks! Helps are greatly appreciated!

Answer 1

Hint: we have $$\langle \alpha'(t),n(t)\rangle=0.$$ Now take the derivative of the above equation with respect to $t$.