Compute Unit Normal Vector: Why Derivative is Orthogonal

In summary, the unit normal vector of a curve is equal to the first derivative of the unit tangent vector divided by the norm of the derivative, for a parametric vector equation with respect to parameter t. This works because the tangent vector is orthogonal to its derivative. However, this is only true if the tangent vector has a constant length.
  • #1
lordkelvin
22
0
The unit normal vector N of a given curve is equal to the first derivative with respect to t of the unit tangent vector T'(t)divided by the norm of T'(t) (For a parametric vector equation of parameter t.)

I realize this works because T(t) is orthogonal to T'(t), but I don't understand why the derivative of the vector T is orthogonal to T itself.

Can anyone explain to me why the derivative of a tangent vector is orthogonal to the tangent vector? Thanks.
 
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  • #2
lordkelvin said:
Can anyone explain to me why the derivative of a tangent vector is orthogonal to the tangent vector? Thanks.

In general it is not true but if the tangent vectors have constant length then the derivative of the length is zero.
 

1. What is a compute unit normal vector?

A compute unit normal vector is a vector that is perpendicular to the surface of a given function or curve at a specific point. It is used to determine the direction of the derivative at that point.

2. Why is the derivative of a compute unit normal vector orthogonal?

The derivative of a compute unit normal vector is orthogonal because it is always perpendicular to the surface at a given point. This means that the derivative of the normal vector is always perpendicular to the tangent vector of the curve or function at that point.

3. How is the compute unit normal vector used in calculus?

The compute unit normal vector is used in calculus to calculate the direction of the derivative at a specific point on a curve or function. It is also used in finding the normal line to a curve, which is important in optimization problems.

4. What is the relationship between the compute unit normal vector and the gradient vector?

The compute unit normal vector and the gradient vector are related in that they are both perpendicular to the surface at a given point. However, the gradient vector is a vector of the rates at which the function changes in each of the coordinate directions, while the compute unit normal vector is a unit vector that points in the direction of the derivative at that point.

5. Can the compute unit normal vector ever be zero?

No, the compute unit normal vector can never be zero. This is because it is a unit vector, meaning it always has a length of 1. If it had a length of 0, it would not be a vector. However, the derivative of the normal vector can be zero, which means that the surface is flat or has a horizontal tangent at that point.

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