Find the equation of the plane passing through the points (1, 1, 0), (1, 2, 1) and (−2, 2, −1).

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Accepted Answer

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> [!TLDR]
> Form two vectors lying in the plane from the three given points, take their cross product to get a normal vector, then write the plane equation using one of the points.
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> Let $A(1,1,0)$, $B(1,2,1)$, $C(-2,2,-1)$.
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> $$\vec{AB} = (1-1,\,2-1,\,1-0) = (0,1,1)$$
> $$\vec{AC} = (-2-1,\,2-1,\,-1-0) = (-3,1,-1)$$
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> Normal vector $\vec n = \vec{AB}	imes\vec{AC}$:
> $$\vec n = \begin{vmatrix}\hat i & \hat j & \hat k\ 0 & 1 & 1\ -3 & 1 & -1\end{vmatrix} = \hat i(1\cdot(-1)-1\cdot1) - \hat j(0\cdot(-1)-1\cdot(-3)) + \hat k(0\cdot1-1\cdot(-3))$$
> $$= \hat i(-2) - \hat j(3) + \hat k(3) = (-2,-3,3)$$
> …

Q.Find the equation of the plane passing through the points (1, 1, 0), (1, 2, 1) and (−2, 2, −1).

Concept understanding — Equation of a Plane

Equation of a Plane

Point + normal form

Normal (perpendicular) form

Intercept form …

Q.Find the equation of the plane passing through the points (1, 1, 0), (1, 2, 1) and (−2, 2, −1).

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