Question

Find the Cartesian equation of the plane \(\vec{r}.(2\hat{i}+\hat{j}-\hat{k})\)=4.

(a) x+y-z=-4

(b) 2x+y-z=4

(c) x+y+z=4

(d) -2x-y+z=4

I have been asked this question in an international level competition.

The origin of the question is Three Dimensional Geometry topic in division Three Dimensional Geometry of Mathematics – Class 12

Answer 1

Correct option is (b) 2x+y-z=4

The best explanation: Given that the equation of the plane is \(\vec{r}.(2\hat{i}+\hat{j}-\hat{k})\)=4

We know that, \(\vec{r}=x\hat{i}+y\hat{j}+z\hat{k}\)

∴\((x\hat{i}+y\hat{j}+z\hat{k}).(2\hat{i}+\hat{j}-\hat{k})\)=4

⇒2x+y-z=4 is the Cartesian equation of the plane.