Explain how the vectors $\mathbf{T}, \mathbf{N}$, and $\mathbf{B}$ are related geometrically. Choose the correct answer below. A. All three vectors are mutually orthogonal at all points of the curve. B. The vectors $\mathbf{T}$ and $\mathbf{N}$ are orthogonal at all points of the curve, but $\mathbf{B}$ is only orthogonal to $\mathbf{T}$ and $\mathbf{N}$ if the curve is two-dimensional. C. The vectors $\mathbf{T}$ and $\mathbf{N}$ are orthogonal at all points of the curve, but $\mathbf{B}$ is only orthogonal to $\mathbf{T}$ and $\mathbf{N}$ if the curve is three-dimensional. D. The vectors $\mathbf{T}, \mathbf{N}$, and $\mathbf{B}$ form a right-handed coordinate system called the osculating plane.
Solution
Step 1 ：The vectors $\mathbf{T}$, $\mathbf{N}$, and $\mathbf{B}$ are known as the Frenet-Serret frame or TNB frame in differential geometry. They are used to describe the local geometric properties of a curve in three-dimensional space.
Step 2 ：$\mathbf{T}$ is the unit tangent vector, which points in the direction of the curve's tangent line.
Step 3 ：$\mathbf{N}$ is the unit normal vector, which points in the direction of the curve's principal normal line.
Step 4 ：$\mathbf{B}$ is the unit binormal vector, which is the cross product of $\mathbf{T}$ and $\mathbf{N}$, and points in the direction that is orthogonal to both $\mathbf{T}$ and $\mathbf{N}$.
Step 5 ：Therefore, all three vectors are mutually orthogonal at all points of the curve, and they form a right-handed coordinate system.
Step 6 ：\(\boxed{\text{The correct answer is A. All three vectors are mutually orthogonal at all points of the curve.}}\)
