# perpendicularity of two vectors in 3D: math problem in a physics context

this is NOT a question about basics of vectors in calculus or geometry. this is NOT a question about dot (inner or scalar) product. This is also not a physics problem even though the context is physics.

here is a derivation from Hallidays physics textbook. but this very derivation comes up in various situation both in mechanics and also in electromagnetism. picture is attached.

we are looking at the cross product of a position vector calleed ri (r sub i) thar points to a differential mass element mi at point P. this mass element mi is going around a circle of a radius equal to the component of ri in the xy plane. please see image please see picture.

at point P we have the vector pi (p sub i). now in the derivation it says that ri and pi are perpendicular so the sin of the angle that they make is 1. I cant wrap my head around it since ri obviously has a component in the xy plane so obvioiusly the two are not perpendicular. am i missing something about the definition of perpendicularity of two vectors?

i tried various ri and pi vectors (with diffrrent numbers) and their dot product never comes out to be zero so these are never really perpendicular going by dot product being zero as the definition of perpendicular.

in one sentence, the question is: why are ri and pi perpendicular?

## 1 Answer

'...since r obviously has a component in the x-y plane so obviously the two are not perpendicular' is where the problem lies. You see, the two vectors have components in the x-y plane, but that does not mean the vectors can't be perpendicular. An elementary example will be to take unit vectors along x-axis(say, x) and along y-axis(say, y). Clearly, x is perpendicular to y, and both vectors very well lie in the x-y plane. However, I'm curious to know how you could obtain non-zero dot products. I'm suspicious you're not allowing the vectors to be perpendicular. As to why they should be perpendicular- p is being drawn tangentially to the circle described by vector r, so for every r, p is naturally perpendicular to it. Although this is MSE, I'll still take some liberty to discuss the physics. You spin a top, say clockwise with it's axis perpendicular to the ground. How do you describe the motion of any particle present in the top? Let's say we talk about an arbitrary point on the surface of the top. The momentum of a particle at that point will be tangential to the circle described by it's position vector(from the axis of rotation) due to the very nature of uniform circular motion. Then from elementary geometry, you do know that tangent to a circle is perpendicular to the radius joining it at the point where it touches the circle