How to proof tangential and normal components of a vector
Thank you
any text on vector analysis or differential geometry will discuss this topic.
And google is your friend.
Couldn't find any proofs for these
Maybe a link would be great]
Thanks
To prove the tangential and normal components of a vector, you need to understand the concept of vector projection.
1. Tangential Component: The tangential component of a vector represents the part of the vector that is parallel to a given curve or path. To find the tangential component of a vector, follow these steps:
a. Determine the unit tangent vector of the curve or path at a particular point. This can be done by finding the derivative of the curve or by using the formula: T = (v / ||v||), where "v" represents the vector.
b. Find the dot product between the vector you want to decompose and the unit tangent vector. The dot product between two vectors measures how much they are aligned.
c. Multiply the dot product by the unit tangent vector. The resulting vector will be the tangential component of the original vector.
2. Normal Component: The normal component of a vector represents the part of the vector that is perpendicular to a given curve or path. To find the normal component, follow these steps:
a. Determine the unit normal vector of the curve or path at a particular point. The unit normal vector can be found by finding the derivative of the vector or by using the formula: N = (dT / ds), where "dT" represents the derivative of the unit tangent vector and "ds" represents the derivative of the curve or path.
b. Find the dot product between the vector you want to decompose and the unit normal vector.
c. Multiply the dot product by the unit normal vector. The resulting vector will be the normal component of the original vector.
By following these steps, you can decompose a vector into its tangential and normal components.