Nearest point to origin of a hyperplane by Lagrange multiplier

Question

Find the point on the hyperplane $x^Tc = β $that is closest to the origin by Lagrange multiplier method.

What is hyperplane and how we obtain its origin ! I need a serious hint !!

Answer 1

HINT: You need solve this problem $$ \min_{x\in D}\frac{1}{2}\|x\|^2,\qquad D=\{x\in\mathbb{R}^n~:~x^{T}c-\beta=0\}. $$ For this, define the Lagrangian by $$ L(x,\lambda)=\frac{1}{2}\|x\|^2-\lambda(x^{T}c-\beta). $$ Imposing the condition $L_{x}(x,\lambda)=0$, we have $x=\lambda c$. As $x$ must belong to $D$ we have that $\lambda=-\beta/\|c\|^2$. So $x=-\frac{\beta c}{\|c\|^2}$. Remark: Check all the details that I did not do.