How does the definition of the vector cross product, using $xyz$ components of two vectors $\mathbf{b}$ and $\mathbf{c}$:
$$\mathbf{b}\times\mathbf{c}=(b_y c_z−b_z c_y,b_z c_x−b_x c_z,b_x c_y−b_y c_x)$$
end up translating into the more geometric $$\Vert \mathbf{b}\times\mathbf{c}\Vert=\Vert \mathbf{b}\Vert\Vert \mathbf{c}\Vert\sin(\theta)$$
where $\theta$ is the angle between $\mathbf{b}$ and $\mathbf{c}$?
Hint: first verify that $(b\times c)^2+(b\cdot c)^2=b^2c^2$ (I'll leave the algebra to you) where $v^2$ is defined as $v\cdot v$ for any vector $v$, so $(b\times c)^2=b^2c^2(1-\cos^2\theta)=(bc\sin\theta)^2$.