Find the equation of the plane passing through the three points (2,2,0), (1,2,1), (-1,2,-2).

Category: QuestionsFind the equation of the plane passing through the three points (2,2,0), (1,2,1), (-1,2,-2).
Editor">Editor Staff asked 11 months ago

Find the equation of the plane passing through the three points (2,2,0), (1,2,1), (-1,2,-2).
 
(a) \((\vec{r}-(2\hat{i}+2\hat{j})).[(-\hat{i}+\hat{k})×(-3\hat{i}-2\hat{k})]\)=0
 
(b) \((\vec{r}-(3\hat{i}-2\hat{k})).[(-\hat{i}+\hat{k})×(2\hat{i}-2\hat{j})]\)=0
 
(c) \((\vec{r}+(2\hat{i}+2\hat{j})).[(-\hat{i}-\hat{k})×(-3\hat{i}-2\hat{k})]\)=0
 
(d) \((\vec{r}-(2\hat{i}+2\hat{j})).[(-\hat{i}-\hat{k})×(3\hat{i}+2\hat{k})]\)=0
 
I have been asked this question during an interview.
 
Origin of the question is Three Dimensional Geometry in portion Three Dimensional Geometry of Mathematics – Class 12
NCERT Solutions for Subject Clas 12 Math Select the correct answer from above options 
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1 Answers
Editor">Editor Staff answered 11 months ago

Correct option is (a) (\vec{r}-(2\hat{i}+2\hat{j})).[(-\hat{i}+\hat{k})×(-3\hat{i}-2\hat{k})]=0
 
For explanation I would say: Let \vec{a}=2\hat{i}+2\hat{j}, \,\vec{b}=\hat{i}+2\hat{j}+\hat{k}, \,\vec{c}=-\hat{i}+2\hat{j}-2\hat{k}
 
The vector equation of the plane passing through three points is given by
 
(\vec{r}-\vec{a}).[(\vec{b}-\vec{a})×(\vec{c}-\vec{a})]=0
 
(\vec{r}-(2\hat{i}+2\hat{j})).[((\hat{i}+2\hat{j}+\hat{k})-(2\hat{i}+2\hat{j}))×((-\hat{i}+2\hat{j}-2\hat{k})–(2\hat{i}+2\hat{j}))]=0
 
(\vec{r}-(2\hat{i}+2\hat{j})).[(-\hat{i}+\hat{k})×(-3\hat{i}-2\hat{k})]=0.