How is T the tangent to a vector

[hivesaeed4]

The tangent vector is defined as :
T=v/||v||
Where v is some vector.
Then how is T the tangent vector to v? It's the unit vector in the direction of v right?

 

[hivesaeed4]

My bad.
v is the velocity of a distance vector.

 

[hivesaeed4]

But even so, if v is the velocity vector I understand how T will be perpendicular to the path of the object as long as its direction is changing like if its moving in a circle. But suppose the object was moving in a straight line. Then would'nt the tangent vector given by the above equations be a unit vector in the same straight line. In other words it would not be a tangent vector. Help?

 

[jppike]

I think you are a bit confused about what it means for something to be tangent. A tangent vector is NOT a normal vector, it should not be perpendicular to the path; it should be "parallel". Recall that a tangent line to a point t=x on the path β(t) is a line that passes through the point (x,β(x)) and has slope β'(x). Hence a line is its own tangent line, so there is no problem with your definition of a tangent vector. Hope that helps