Can someone tell me how they did cross product?

[Warlic]

Homework Statement

Angular momentum is the cross product of r and mv. But why is there mvR outside of the paranthesis? And where did the v go in the second paranthesis - shouldn't the second paranthesis be (-v*sin(ωt), v* cos(ωt)). Does anyone have any idea how they did the cross product, because I'm totally lost.

2. Homework Equations

The Attempt at a Solution

 

[Molar]

Go through how vector cross products are done :
https://en.wikipedia.org/wiki/Cross_product

Also, m,v, R - all being scalar ( just numbers, see they are without arrows) are multiplied dierctly and so kept outside while the main (vector) components of cross product is kept within the third bracket.

Second paranthesis, v/R is replaced by angular velocity "ω".

See the link for the process.

 

[Fredrik]

The cross product is bilinear. That means that for all vectors $\vec x,\vec y,\vec z$ and all numbers $a,b$, we have
$$(a\vec x+b\vec y)\times \vec z =a(\vec x\times\vec z) +b(\vec y\times\vec z)$$ and
$$\vec x\times (a\vec y+b\vec z)=a(\vec x\times y)+b(\vec x\times\vec z).$$