Question

Consider a point given by position vector $\overline{a}.$ Distance of this point from the plane given $\overline{r}.\overline{n}=d$ will be.

• A. $\frac{\overline{a}.\overline{n}+d}{|\overline{n}|}$
• B. $\frac{\overline{a}.\overline{n}-d}{|\overline{n}|}$
• C. $\frac{\overline{a}.\overline{n}}{|\overline{n}|}$
• D. $Noneofthese$

Answer 1

The correct option is B $\frac{\overline{a}.\overline{n}-d}{|\overline{n}|}$

We know the direction vector $\hat{n}$ will be passing through the origin. If we calculate the projection of the given point on the plane along this vector $\hat{n}$ then subtract the distance of the plane from origin we get the perpendicular distance between point and plane. We already know distance of plane from origin will be $\frac{\overline{r}.\overline{n}}{|\overline{n}|}=\frac{d}{|\overline{n}|}$. Projection of given position vector will be $\frac{\overline{a}.\overline{n}}{|\overline{n}|}$. $\therefore$ length of perpendicular will be $\frac{\overline{a}.\overline{n}-d}{|\overline{n}|}$