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Which of the following is an equation for the plane through the point (1, 1, −1) and parallel

to the plane x − y + z = 3? Show all your work.

Which of the following is an equation for the plane through the point (1, 1, −1) and-example-1
User Warpech
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1 Answer

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The given plane,
x-y+z=3, has normal vector
\vec n = \langle1,-1,1\rangle. Any plane parallel to this one has the same normal vector.

Let
(x,y,z) be any point in the plane we want. The plane contains the point (1, 1, -1), so an arbitrary vector in this plane is


\langle x,y,z\rangle - \langle 1,1,-1\rangle = \langle x-1, y-1, z+1 \rangle

and this is perpendicular to
\vec n.

So the equation of the plane is


\langle x-1, y-1, z+1 \rangle \cdot \vec n = 0 \implies (x-1) - (y-1) + (z+1) = 0 \implies \boxed{x - y + z = -1}

or equivalently,


\boxed{-x + y - z = 1}

User TheCrafter
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