Question

The vector equation of the plane through the point $2\mathbf{i} - \mathbf{j} - 4\mathbf{k}$ and parallel to the plane $\mathbf{r}.(4\mathbf{i} - 12\mathbf{j} - 3\mathbf{k}) - 7 = 0$ is

• A. $\mathbf{r}.(4\mathbf{i} - 12\mathbf{j} - 3\mathbf{k}) = 0$
• B. $\mathbf{r}.(4\mathbf{i} - 12\mathbf{j} - 3\mathbf{k}) = 32$
• C. $\mathbf{r}.(4\mathbf{i} - 12\mathbf{j} - 3\mathbf{k}) = 12$
• D. None of these

Answer 1

Answer: (B)

$\mathbf{r}.(4\mathbf{i} - 12\mathbf{j} - 3\mathbf{k}) = 32$

Answer 2

The equation of a plane parallel to the plane

$\mathbf{r}.(4\mathbf{i} - 12\mathbf{j} - 3\mathbf{k}) - 7 = 0$is $\mathbf{r}.(4\mathbf{i} - 12\mathbf{j} - 3\mathbf{k}) + \lambda = 0$.

This passes through$2\mathbf{i} - \mathbf{j} - 4\mathbf{k}$.

Therefore,$(2\mathbf{i} - \mathbf{j} - 4\mathbf{k}).(4\mathbf{i} - 12\mathbf{j} - 3\mathbf{k}) + \lambda = 0$

⇒ $8 + 12 + 12 + \lambda = 0 \Rightarrow \lambda = - 32$

So, the required plane is $\mathbf{r}.(4\mathbf{i} - 12\mathbf{j} - 3\mathbf{k}) - 32 = 0$.