Question

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

Answer: (C)

Are on the opposite side of the plane

Answer 2

The position vectors of two given points are $\mathbf{a} = \mathbf{i} - \mathbf{j} + 3\mathbf{k}$ and $\mathbf{b} = 3\mathbf{i} + 3\mathbf{j} + 3\mathbf{k}$ the equation of the given plane is $\mathbf{r}.(5\mathbf{i} + 2\mathbf{j} - 7\mathbf{k}) + 9 = 0$ or $\mathbf{r}.\mathbf{n} + d = 0$.

We have, $\mathbf{a}.\mathbf{n} + d = (\mathbf{i} - \mathbf{j} + 3\mathbf{k}).(5\mathbf{i} + 2\mathbf{j} - 7\mathbf{k}) + 9$

$= 5 - 2 - 21 + 9 < 0$

and, $\mathbf{b}.\mathbf{n} + d = (3\mathbf{i} + 3\mathbf{j} + 3\mathbf{k}).(5\mathbf{i} + 2\mathbf{j} - 7\mathbf{k}) + 9$

$= 15 + 6 - 21 + 9 > 0$

So, the points a and $\mathbf{b}$are on the opposite sides of the plane.