The Binormal vector, B, is defined as B = T x N where T and N are the unit tangent and unit normal, respectively. What is the value of T.TxB and B.TxN?

The dot product of any vector with itself is always a positive number, except for the zero vector, which is 0. However, in this case, T and B are perpendicular to each other by definition (since B is defined as the cross product of T and N, and the cross product of any two vectors is always perpendicular to both). Therefore, the dot product of T and B (T.B) is 0.

For the second part, B.TxN, we know that B = T x N. So, B.TxN becomes (T x N).T x N. The dot product is distributive over the cross product, so this can be rewritten as T.T x N - T x N.N. But the dot product of any vector with itself (T.T or N.N) is a positive number, and the cross product of any vector with itself (T x T or N x N) is the zero vector. Therefore, B.TxN = 0 - 0 = 0.