Proving the orthogonality of curvature and unit tangent

Question

Consider the unit tangent $T = P'(t)/||P'(t)||$ and curvature $K = T'$ of a curve $P(t)$. The curvature being the derivative of the unit tangent by definition.

If I were to show that $K$ is orthogonal to $T$, would it be mathematically enough to show that

$2T * T' = 0$

derived from the fact that the dot product of T with itself is one. It is quite intuitive that if $2T$ is orthogonal to something then $T$ must be as well, but is that a solid proof? If not, then how would I show the orthogonality of K and T?

Answer 1

Yes, that is correct.

Just divide by 2.

Answer 2

A scalar multiple of a vector has the same direction as the unscaled vector when the scalar is positive, and opposite direction when the scalar is negative. So if $2T \cdot T' = 0$, then $2T$ and $T'$ are orthogonal - which means that $T$ and $T'$ are orthogonal by the first sentence.