Say I'm taking the cross product of vectors $a$ and $b$. Say that $b$ is totally in the $z$ direction and has length $7$, so $b = 7k$. Say that $a$ is in the $xy$-plane with positive coefficients, $a = 3x + 4y$.
I want to understand the sign of the components of $a \times b$ using the right hand rule. Now, surele since $a \times b$ is orthogonal to both $a$ and $b$, it's $z$ component will be zero. But will the $x$ and $y$ components be positive or negative, and how can i see this with the right hand rule? Thank you for your time