Question

The vector equation of a plane, which is at a distance of 8 unit from the origin and which is normal to the vector $2\mathbf{i} + \mathbf{j} + 2\mathbf{k},$ is

• A. $\mathbf{r}.(2\mathbf{i} + \mathbf{j} + \mathbf{k}) = 24$
• B. $\mathbf{r}.(2\mathbf{i} + \mathbf{j} + 2\mathbf{k}) = 24$
• C. $\mathbf{r}.(\mathbf{i} + \mathbf{j} + \mathbf{k}) = 24$
• D. None of these

Answer 1

Answer: (B)

$\mathbf{r}.(2\mathbf{i} + \mathbf{j} + 2\mathbf{k}) = 24$

Answer 2

Here $d = 8$ and $\mathbf{n} = 2\mathbf{i} + \mathbf{j} + 2\mathbf{k})$

∴ $\widehat{\mathbf{n}} = \frac{\mathbf{n}}{|\mathbf{n}|} = \frac{2\mathbf{i} + \mathbf{j} + 2\mathbf{k}}{\sqrt{4 + 1 + 4}} = \frac{2}{3}\mathbf{i} + \frac{1}{3}\mathbf{j} + \frac{2}{3}\mathbf{k}$

Hence, the required equation of the plane is

$\mathbf{r}.\left( \frac{2}{3}\mathbf{i} + \frac{1}{3}\mathbf{j} + \frac{2}{3}\mathbf{k} \right) = 8$ or $\mathbf{r}.(2\mathbf{i} + \mathbf{j} + 2\mathbf{k}) = 24$.