Question

The equation of the plane passing through three non-collinear points with position vectors $\vec{a}$, $\vec{b}$, $\vec{c}$ is

• A. $\vec{r}\cdot \left(\vec{b}\times \vec{c}+\vec{c}\times \vec{a}+\vec{a}\times \vec{b}\right)=0$
• B. $\vec{r}\cdot \left(\vec{b}\times \vec{c}+\vec{c}\times \vec{a}+\vec{a}\times \vec{b}\right)=\left[\vec{a}\vec{b}\vec{c}\right]$
• C. $\vec{r}\cdot \left(\vec{a}\times \left(\vec{b}+\vec{c}\right)\right)=\left[\vec{a}\vec{b}\vec{c}\right]$
• D. $\vec{r}\cdot \left(\vec{a}+\vec{b}+\vec{c}\right)=0$

Answer 1

Answer: (B)

Let $P\left(\vec{r}\right)$ be any point on plane.
Clearly $\vec{r}-\vec{a}$ will be in linear combination of $\vec{b}-\vec{a}$ and $\vec{c}-\vec{a}$
$\Rightarrow \vec{r}-\vec{a}$,
$\vec{b}-\vec{a}$,
$\vec{c}-\vec{a}$ will be coplanar
$\Rightarrow \left(\vec{r}-\vec{a}\right)\cdot \left\{\left(\vec{b}-\vec{a}\right)\times \left(\vec{c}-\vec{a}\right)\right\}=0$
$\Rightarrow \left(\vec{r}-\vec{a}\right)\cdot \left\{\vec{b}\times \vec{c}+\vec{a}\times \vec{b}+\vec{c}\times \vec{a}\right\}=0$
$\Rightarrow \vec{r}\cdot \left\{\vec{b}\times \vec{c}+\vec{c}\times \vec{a}+\vec{a}\times \vec{b}\right\}=\left[\vec{a}\vec{b}\vec{c}\right]$