# Find an equation of the plane through the origin and the points (3, -2, 1) and (1, 1, 1).

## Question:

Find an equation of the plane through the origin and the points {eq}(3, -2, 1) {/eq} and {eq}(1, 1, 1) {/eq}.

## Equation Of Plane:

Equation of plane passing through three points can be found by setting up the determinant as shown {eq}\begin{vmatrix} x-x_{1} &y-y_{1} &z-z_{1} \\ x_{2}-x_{1}&y_{2}-y_{1} &z_{2}-z_{1} \\ x_{3}-x_{1}&y_{3}-y_{1} & z_{3}-z_{1} \end{vmatrix}=0\\ {/eq}

## Answer and Explanation:

{eq}\begin{vmatrix} x-x_{1} &y-y_{1} &z-z_{1} \\ x_{2}-x_{1}&y_{2}-y_{1} &z_{2}-z_{1} \\ x_{3}-x_{1}&y_{3}-y_{1} & z_{3}-z_{1} \end{vmatrix}=0\\ \begin{vmatrix} x & y & z\\ 3 & -2 & 1\\ 1& 1& 1 \end{vmatrix}=0\\ -3x-2y+5z=0 {/eq}

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from Precalculus: High School

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