$$\vec{A}=\vec{A}_xi+\vec{A}_yj+\vec{A}_zk$$ $$A^2=\mathrm{A}_{x}^{2}+\mathrm{A}_{y}^{2}+\mathrm{A}_{z}^{2}$$ $$\vec{A}\cdot \vec{B}=AB\cos\theta$$ $$\vec{A}\cdot \vec{B}=A_xB_x+A_yB_y+A_zB_z$$ $$\left|\vec{A}\times\vec{B}\right|=AB\sin\theta$$ $$\vec{A}\times\vec{B}=(A_yB_z-B_yA_z)\hat{i}+(A_zB_x-B_zA_x)\hat{j}+(A_xB_y-B_xA_y)\hat{k}$$