3D Geometry
Plane Equations
MMTS_Full_Test_07
Grade 12
Question:
The vector equation of the plane through the line of intersection of the planes $x+y+z=1$ and $2x+3y+4z=5$ which is perpendicular to the plane $x-y+z=0$ is
$\vec{r}\times(\hat{i}+\hat{k})+2=0$
$\vec{r}\cdot(\hat{i}-\hat{k})-2=0$
$\vec{r}\cdot(\hat{i}-\hat{k})+2=0$
$\vec{r}\times(\hat{i}-\hat{k})+2=0$
Step-by-Step Solution
Key Concept: Family of planes: $(x+y+z-1)+\lambda(2x+3y+4z-5)=0$; find $\lambda$ from perpendicularity
Normal: $(1+2\lambda,1+3\lambda,1+4\lambda)\cdot(1,-1,1)=0\Rightarrow 1+2\lambda-1-3\lambda+1+4\lambda=0\Rightarrow 3\lambda+1=0\Rightarrow\lambda=-1/3$. Plane: $x-z+2=0$. $\vec{r}\cdot(\hat{i}-\hat{k})+2=0$.
Correct Answer: 3