Unit vector cross products in different co-ords

[Jesssa]

hey,

i've been trying to work out how to determine the sign of cross products of unit vectors,

for example in cylindrical,

r x z = - theta

theta x z = r

r x theta = z

i can't figure out the sign,

r x z = |r||z|sinβ theta where β is the angle between them, which is 90°,

and the length of the vectors are 1

how can you tell that its actually -theta?

 

[BruceW]

I can think of 3 different ways to figure this out. You are probably familiar with at least one of them. When you have a cross-product between two vectors, how do you usually work out the direction of the resulting vector?

 

[chiro]

Hey Jesssa and welcome to the forums.

Consider that a x b = |a||b|sin(a,b)N where N is the normal vector and also <a,b> = |a||b|cos(a,b) where a x b is the cross product and <a,b> is the dot or inner product for Cartesian three dimensional space.

Now consider what sin(a,b) and cos(a,b) should be (in terms of sign) for the various quadrants.

 

[tiny-tim]

i remember it by thinking that θ is in the same direction as y, and then using x x y = z etc