Medium Practice — Set 2

Quick Recap — Planes in Depth

• Distance from $(x_1,y_1,z_1)$ to $ax+by+cz+d=0$: $\dfrac{|ax_1+by_1+cz_1+d|}{\sqrt{a^2+b^2+c^2}}$.
• Parallel planes: normals proportional; distance $=\dfrac{|d_1-d_2|}{\sqrt{a^2+b^2+c^2}}$ (after matching coefficients).
• Perpendicular planes: $\vec n_1\cdot\vec n_2=0$; angle between planes $=$ angle between normals.
• Family of planes through the intersection of $P_1,P_2$: $P_1+\lambda P_2=0$.