Question

The position vectors of point $A$ and $B$ are $\hat{i}-\hat{j}+3\hat{k}$ and $3\hat{i}+3\hat{j}+3\hat{k}$ respectively. The equation of a plane is $r.(5\hat{i}+2\hat{j}-7\hat{k})+9=0$. The point $A$ and $B$

• A. lie on the plane
• B. are on the same side of the plane
• C. are on the opposite side of the plane
• D. None of these

Answer 1

The correct option is C are on the opposite side of the plane

The position vectors of two given points are $a=\hat{i}-\hat{j}+3\hat{k}$ and $b=3\hat{i}+3\hat{j}+3\hat{k}$ and the equation of the given plane is $r.(5\hat{i}+2\hat{j}+7\hat{k})+9=0$ or $r.n+d=0.$
We have, $a.n.+d=(\hat{i}-\hat{j}-\hat{k}).(5\hat{i}+2\hat{j}-7\hat{k})+9=5-2-21+9<0$
and $b.n+d=(3\hat{i}+3\hat{j}+3\hat{k}).(5\hat{i}+2\hat{j}+7\hat{k})+9=15+6-21+9>0$
So, the points $a$ and $b$ are on the opposite sides of the plane.