# Vector – Equation of a Plane

## Vector – Equation of a Plane

Equation of the Plane that Passes through Point A with Position Vector $$\vec{a}$$ and is Parallel to given Vector $$\vec{b}$$ and $$\vec{c}$$:

Vector form: Let $$\vec{r}$$ be the position vector of any Point P in the plane.

Then $$\overrightarrow{AP}=\overrightarrow{OP}-\overrightarrow{OA}=\vec{r}-\vec{a}$$,

Since vectors $$\vec{r}-\vec{a}$$, $$\vec{b}$$ and  $$\vec{c}$$ are coplanar,

we have $$\left( \vec{r}-\vec{a} \right).\left( \vec{b}\times \vec{a} \right)=0$$,

$$\vec{r}.\left( \vec{b}\times \vec{a} \right)-\vec{a}.\left( \vec{b}\times \vec{a} \right)=0$$,

$$\vec{r}.\left( \vec{b}\times \vec{a} \right)=\vec{a}.\left( \vec{b}\times \vec{a} \right)$$,

$$\left[ \overrightarrow{r}\,\,\overrightarrow{b}\,\,\overrightarrow{c} \right]=\left[ \overrightarrow{a}\,\,\overrightarrow{b}\,\,\overrightarrow{c} \right]$$,

Which is the required equation of the plane.

Cartesian Form: From $$\left( \vec{r}-\vec{a} \right).\left( \vec{b}\times \vec{a} \right)=0$$,

We have $$\left[ r-\overrightarrow{a}\,\,\overrightarrow{b}\,\,\overrightarrow{c} \right]$$,

$$\left[ \begin{matrix} x-{{x}_{1}} & y-{{y}_{1}} & z-{{z}_{1}} \\ {{x}{2}} & {{y}{2}} & {{z}{2}} \\ {{x}{3}} & {{y}{3}} & {{z}_{3}} \\\end{matrix} \right]=0$$,

Which is the required equation of the plane

Where $$\vec{b}={{x}_{2}}\hat{i}+{{y}_{2}}\hat{j}+{{z}_{2}}\hat{k}$$ and $$\vec{c}={{x}_{3}}\hat{i}+{{y}_{3}}\hat{j}+{{z}_{3}}\hat{k}$$.