Question

A distância do ponto P (2, 1,-1) ao plano a: 2x +3y-4z+1=0 é igual a:​

Answer 1

$\Large\boxed{\begin{array}{l}\underline{\rm Dist\hat ancia~de~um~ponto~P_0(x_0,y_o,z_0)}\\\underline{\rm ao\,plano}\\\underline{\rm\pi:ax+by+cz+d=0}\\\\\sf d(P_0,\pi)=\dfrac{|ax_0+by_0+cz_0+d|}{\sqrt{a^2+b^2+c^2}}\end{array}}$

$\Large\boxed{\begin{array}{l}\sf a:2x+3y-4z+1=0\\\sf P=(2,1,-1)\\\\\sf d(P,a)=\dfrac{|2\cdot2+3\cdot1-4\cdot(-1)+1|}{\sqrt{2^2+3^2+(-4)^2}}\\\\\sf d(P,a)=\dfrac{|4+3+4+1|}{\sqrt{4+9+16}}\\\\\sf d(P,a)=\dfrac{12}{\sqrt{29}}\cdot\dfrac{\sqrt{29}}{\sqrt{29}}\\\\\sf d(P,a)=\dfrac{12\sqrt{29}}{29}\end{array}}$