Questions concerning cross products, dot products, and polar coordinates

[don_anon25]

Question answered! Thanks!

 

[sic]

Actually it is simplier to begin from (2)
Explanation of the terms: first term is the vector parallel to e, second orthogonal to e
A x e is normal to both A and e and has magnitude of the projection of A on the plane normal to e, and e x ( A x e ) has the same magnitude and is normal to e and previous vector..
(1): express X as shown in (2)
(3): pay attention that the curve is a circle (try to make sure..).. as you probably know, the force applied on a mass rotating with a constant speed is normal to the direction of its movement (and so is acceleration), so the dot product is zero..
about derivations in polar coordinates: locally, at any point you have ortogonal axes r and theta, so locally there is no difference between cartesian and polar coordinates, and that's what you need to know for finding derivatives. making dot and cross products in spherical coordinates is not nice (but there are formulas for that), so it is better to go though cartesian coordinates.

 

[quicknote]

Cross products and dot products are mathematical operations that are commonly used in vector algebra. The cross product of two vectors is a vector that is perpendicular to both of the original vectors and has a magnitude equal to the product of their magnitudes multiplied by the sine of the angle between them. This operation is useful for calculating the area of a parallelogram formed by two vectors, as well as determining the direction of rotation between two vectors.

The dot product, on the other hand, is a scalar quantity that is calculated by multiplying the magnitudes of two vectors and the cosine of the angle between them. This operation is useful for determining the angle between two vectors and for projecting one vector onto another.

Polar coordinates are a way of representing points in a two-dimensional space using a distance (r) from the origin and an angle (θ) from a reference axis. This system is useful for describing circular or rotational motion, as well as for simplifying certain mathematical equations.

I hope this helps clarify the concepts of cross products, dot products, and polar coordinates. If you have any further questions, please don't hesitate to ask.