Tangent Vectors

A tangent vector is tangent to a curve or a surface or something.

Most definitions you read will talk about going through a single point on that curve or Surface or something. We could make this slightly more difficult in two ways (see Appendix B).

If we draw a circle and pick a point on it, we can draw one line that is tangent through that point.

If we draw a sphere and pick a point on it, then the infinite set of lines that we can draw through that point build a tangent plane.

Appendix A

For the first example, all acceptable tangent vectors coincide with that line.

We might choose to “build” a space using all possible tangent vectors– we end up building that line.

We could continue but we are probably going too far if we start talking about tangent spaces.

Appendix B

If our function is a line, then the tangent vector is going to go through all the poiby of the function.

If our function has enough curviness (we will speak of the function’s Concavity if we wish for more technicality), it can go in one direction past the point of tangency and later change to the other direction and if it goes far enough, without changing again, it will intersect the tangent line a second time.

We use our intelligence to judge that this is allowable.