Exercise 858Calculus › Multivariable calculus › Optimisation in several variablesL1Added 3 Nov 2026

Distance from point to plane

#Lagrange#distance#plane

Statement

Find the shortest distance from $(1,1,1)$ to the plane $x+y+z=0$ .

Hint

Minimize $d^2=(x-1)^2+(y-1)^2+(z-1)^2$ under $x+y+z=0$ .

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Lagrange. $2(x-1)=\lambda$ , $2(y-1)=\lambda$ , $2(z-1)=\lambda$ , so $x=y=z$ . Constraint: $3x=0$ , so $x=y=z=0$ .

Distance. $\sqrt{(1)^2+(1)^2+(1)^2}=\sqrt3$ .

Conclusion. Distance $=\sqrt3$ .