Question

The length of the perpendicular from the origin to the plane passing through the point $\overline{a}$ and containing the line $\overline{r}=\overline{b}+\lambda \overline{c}$ is :

• A. $=\frac{\left[\overline{a}\overline{b}\overline{c}\right]}{|\overline{a}\times \overline{b}+\overline{b}\times \overline{c}|}$
• B. $=\frac{\left[\overline{a}\overline{b}\overline{c}\right]}{|\overline{b}\times \overline{c}+\overline{c}\times \overline{a}|}$
• C. $=\frac{\left[\overline{a}\overline{b}\overline{c}\right]}{|\overline{c}\times \overline{a}+\overline{a}\times \overline{b}|}$
• D. $=\frac{\left[\overline{a}\overline{b}\overline{c}\right]}{|\overline{c}\times \overline{a}-\overline{a}\times \overline{b}|}$

Answer 1

The correct option is B $=\frac{\left[\overline{a}\overline{b}\overline{c}\right]}{|\overline{b}\times \overline{c}+\overline{c}\times \overline{a}|}$

Plane passes through the point $\overline{a}$ and contains plane $\overline{r}=\overline{b}+\lambda \overline{c}$
$\therefore$ The plane contains the points $\overline{a}$ and $\overline{b}$ and is parallel to the vector $\overline{c}$.
Let $\overline{r}$ be any point in the plane. Then $\overline{r}-\overline{a},\overline{b}-\overline{a}$ and $\overline{c}$ are coplanar.
$\therefore$ Equation of plane is $[\overline{r}-\overline{a},\overline{b}-\overline{a},\overline{c}]=0$
i.e., $\overline{r}.(\overline{b}\times \overline{c}+\overline{c}\times \overline{a})=\left[\overline{a}\overline{b}\overline{c}\right]$
Hence, $\perp$ distance of the plane from the origin
$=\frac{\left[\overline{a}\overline{b}\overline{c}\right]}{|\overline{b}\times \overline{c}+\overline{c}\times \overline{a}|}$