Maths Olympiad Prep

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Geometry Difficulty 5.1 AIME, harder Find the answer

## [Coordinate method in space ]

[ equation of a plane ]]

Form the equation of the plane passing through the point M0(x0;y0;z0)M 0(x 0 ; y 0 ; z 0) perpendicular to a non-zero vector n⃗\vec{n} =(a;b;c)=(a ; b ; c).

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Solution

Point M(x;y;z)M(x ; y ; z) belongs to the desired plane α\alpha if and only if the vector M0M→=(x−x0;y−y0;z−z0)\overrightarrow{M_{0} M}=(x-x 0 ; y-y 0 ; z-z 0) is perpendicular to vector n⃗⋅\vec{n} \cdot Therefore,

M(x;y;z)∈α⇔n⃗⊥M0M→⇔n⃗⋅M0M→=0⇔⇔a(x−x0)+b(y−y0)+c(z−z0)=0 \begin{aligned} & M(x ; y ; z) \in \alpha \Leftrightarrow \vec{n} \perp \overrightarrow{M_{0} M} \Leftrightarrow \vec{n} \cdot \overrightarrow{M_{0} M}=0 \Leftrightarrow \\ & \Leftrightarrow a(x-x 0)+b(y-y 0)+c(z-z 0)=0 \end{aligned}

## Answer

a(x−x0)+b(y−y0)+c(z−z0)=0a(x-x 0)+b(y-y 0)+c(z-z 0)=0

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Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic and difficulty added by this site.