- #1

- 30

- 0

I was reading about the tangent vector at a point on a curve.

It is formulated as

where

All I can infer from the formulation is that the tangent is the rate of change of a position vector at a point in question. But then a tangent vector cannot have its components to be rates of change. Because then the defined tangent vector is the rate of change of tangent vector and not the tangent vector itself.

Therefore I think that the tangent vector should be formulated as:

Lim Δt→0 [

This would have both the direction and magnitude at a point (or in a very very small interval in this case as Δt→0 ). This could also give us the equation of the tangent at the point.

My question is simple. Why would we represent a vector (tangent vector) at a point by the rate of change of position vector at the point? Shouldn't it just be the difference between two position vectors with Δt→0?

Please tell me if what I think of the formulation is correct.

Thanks

It is formulated as

**r'**= Lim Δt→0 [**r**(t+Δt) -**r**(t)] / Δt (sorry for the misrepresentation of the 'Lim Δt→0 ')where

**r**(t) is a position vector to the curve and t is a parameter and**r'**is the derivative of**r**(t).All I can infer from the formulation is that the tangent is the rate of change of a position vector at a point in question. But then a tangent vector cannot have its components to be rates of change. Because then the defined tangent vector is the rate of change of tangent vector and not the tangent vector itself.

Therefore I think that the tangent vector should be formulated as:

Lim Δt→0 [

**r**(t+Δt) -**r**(t)].This would have both the direction and magnitude at a point (or in a very very small interval in this case as Δt→0 ). This could also give us the equation of the tangent at the point.

My question is simple. Why would we represent a vector (tangent vector) at a point by the rate of change of position vector at the point? Shouldn't it just be the difference between two position vectors with Δt→0?

Please tell me if what I think of the formulation is correct.

Thanks

Last edited: