21. As an example suppose X is a timelike tangent vector with respect to the \, g metric. 22. Thus the length of a tangent vector cannot be defined, but the angle between two vectors still can. 23. Is a tangent vector to " C " in curvilinear coordinates ( using the chain rule ). 24. In modern terminology virtual displacement is a tangent vector to the manifold representing the constraints at a fixed time. 25. There is an associated notion of Carath�odory length for tangent vectors to the ball " B ". 26. More strictly this defines an affine tangent space, distinct from the space of tangent vectors described by modern terminology. 27. To get the angle, we need the tangent vectors and at along the directions of sides and, respectively. 28. Therefore, all tangent vectors in a point p span a linear space, called the tangent space at point p. 29. This means that the directional derivative depends only on the tangent vector of the curve at " p ". 30. Though, for a curve whose tangent vectors all have the same sign of magnitude, the arc length is defined.
