No.jaejoon89 said:Hi, this is a calc 3 question. I know the binormal is given by
B = T x N
where
Binormal vector, B
Tangent vector, T
Normal vector, N
Also,
B = (R'(t) x R''(t)) / | R'(t) x R''(t) |
T = R'(t) / | R'(t) |
Does that mean
N = R''(t) / |R''(t)|?
Binormal and normal vectors are both types of tangent vectors that can be used to describe a curve or a surface in three-dimensional space. The main difference between them is their orientation. Normal vectors are perpendicular to the surface, while binormal vectors are perpendicular to both the surface and the curve. In other words, normal vectors lie on the surface, while binormal vectors lie on the plane that contains the curve and is tangent to the surface at a given point.
In calculus, binormal and normal vectors are used to find the curvature and torsion of a curve or a surface. The curvature measures how much a curve deviates from a straight line, while the torsion measures how much a curve twists in three-dimensional space. Both of these quantities are important in understanding the behavior of a curve or a surface and are calculated using binormal and normal vectors.
To find the binormal vector of a curve, you first need to find the tangent vector and the normal vector at a given point on the curve. The binormal vector can then be calculated by taking the cross product of the tangent and normal vectors. This will result in a vector that is perpendicular to both the tangent and normal vectors, and therefore, perpendicular to the curve and the surface at that point.
Yes, binormal and normal vectors can be used to describe any smooth curve or surface in three-dimensional space. However, if the curve or surface has sharp corners or edges, the concept of curvature and torsion may not be well-defined, and the use of binormal and normal vectors may not be applicable.
Binormal and normal vectors can be visualized as arrows or lines that are perpendicular to the curve or surface at a given point. On a graph, the tangent vector can be represented as a line tangent to the curve, the normal vector can be represented as a line perpendicular to the tangent vector, and the binormal vector can be represented as a line perpendicular to both the tangent and normal vectors.