গোলকের সাথে স্পর্শক তল

For a sphere in $\mathbb{R}^3$ with center $c=(c_1, c_2, c_3)$, and some point on the surface of the sphere $r=(r_1, r_2, r_3)$, define vector $\vec{a} =r-c$. A planes passing point $r$ and tangent with the sphere will be perpendicular to the $\vec{a}$. Using the fact that if the dot product between two vectors equals to zero then the two vectors are perpendicular with each other, we know that if point $p = (p_1, p_2, p_3)$ in $\mathbb{R}^3$ is on the tangent plane, let $\vec{b} = p-c$, then $\vec{a}\cdot\vec{b} = 0$, i.e., $(p_1-c_1)(r_1-c_1)+(p_2-c_2)(r_2-c_2)+(p_3-c_3)(r_3-c_3)=0$. This is the formula for tangent planes to spheres.