So far we have associated two unit vectors with parameterized curves - the unit tangent and the unit normal. There is a third vector known as the [b][color=#ff0000]binormal[/color][/b] defined as the cross product of the unit tangent and the unit normal:[br][br][math]\vec{B}=\vec{T}\times\vec{N}[/math]
To define the unit tangent and unit normal it was required we divide by a length to achieve a unit vector. However in my definition of the binormal there is no such requirement. Explain why the binormal vector must always be a unit vector.
[math]\left|\left|\vec{u}\times\vec{v}\right|\right|=\left|\left|\vec{u}\right|\right|\left|\left|\vec{v}\right|\right|\sin\theta[/math][br]Since [math]\vec{T}[/math] and [math]\vec{N}[/math] are perpendicular unit vectors, their cross product necessarily has length 1.
[math]\vec{B}[/math] is perpendicular to both [math]\vec{T}[/math] and [math]\vec{N}[/math]. We can think of the vectors [math]\vec{T},\vec{N},\text{ and }\vec{B}[/math] as forming a mini-basis for [math]\mathbb{R}^3[/math] where we've re-centered the origin to be the point along the curve. Together these three vectors are called the [b][color=#ff0000]Frenet Frame[/color][/b].