Equation of Plane Passing through Three Points

Q. The minimum number of non collinear points required to define a plane is 3.

1. True
2. False

Q. Given 3 points given by position vectors $\overline{a},\overline{b}$ and $\overline{c}$. The plane which passes through these 3 points can be given by

1. 
2. 
3. 
4. 

Q. Given 3 points given by position vectors $\overline{a}=\hat{i}+\hat{j},\overline{b}=\hat{j}+\hat{k},\overline{c}=-\hat{i}-\hat{j}-\hat{k}$. Find the plane passing through these 3 points.

1. 
2. 
3. 
4. 

Q. Given 3 points given by position vectors $\overline{a}=\hat{i}+\hat{j},\overline{b}=\hat{j}+\hat{k},\overline{c}=-\hat{i}-\hat{j}-\hat{k}$. Find the plane passing through these 3 points.

1. $\overline{r}.(\overline{i}-3\hat{j}+\hat{k})=-1$
2. $\overline{r}.(-\overline{i}+3\hat{j}-\hat{k})=1$
3. $\overline{r}.(\overline{i}-3\hat{j}+\hat{k})=2$
4. $\overline{r}.(-\overline{i}+3\hat{j}-\hat{k})=-2$

Q. In $R^3$, consider the planes $P_1:y=0$ and $P_2:x+z=1.$ Let $P_3$ be a plane, different from $P_1$ and $P_2$, which passes through the intersection of $P_1$ and $P_2.$ If the distance of the point $(0,1,0)$ from $P_3$ is $1$ and the distance of a point $(\alpha ,\beta ,\gamma )$ from $P_3$ is $2$, then which of the following relations is/are true?

1. $2\alpha +\beta +2\gamma +2=0$
2. $2\alpha -\beta +2\gamma +4=0$
3. $2\alpha +\beta -2\gamma -10=0$
4. $2\alpha -\beta +2\gamma -8=0$

Q. The minimum number of non collinear points required to define a plane is 3.

1. True
2. False

Q. Given 3 points given by position vectors $\overline{a}=\hat{i}+\hat{j},\overline{b}=\hat{j}+\hat{k},\overline{c}=-\hat{i}-\hat{j}-\hat{k}$. Find the plane passing through these 3 points.

1. $\overline{r}.(\overline{i}-3\hat{j}+\hat{k})=-1$
2. $\overline{r}.(-\overline{i}+3\hat{j}-\hat{k})=1$
3. $\overline{r}.(\overline{i}-3\hat{j}+\hat{k})=2$
4. $\overline{r}.(-\overline{i}+3\hat{j}-\hat{k})=-2$

Q. Given 3 points given by position vectors $\overline{a},\overline{b}$ and $\overline{c}$. The plane which passes through these 3 points can be given by

1. $(\overline{r}-\overline{a}).\left(\right(\overline{a}+\overline{b})\times (\overline{b}+\overline{c}\left)\right)=0$
2. $(\overline{r}-\overline{a}).\left(\right(\overline{a}+\overline{b})\times (\overline{b}-\overline{c}\left)\right)=0$
3. $(\overline{r}-\overline{b}).\left(\right(\overline{a}-\overline{b})\times (\overline{b}-\overline{c}\left)\right)=0$
4. $(\overline{r}-\overline{a}).\left(\right(\overline{a}-\overline{b})\times (\overline{b}-\overline{c}\left)\right)=0$

Q. Given 3 points given by position vectors $\overline{a},\overline{b}$ and $\overline{c}$. The plane which passes through these 3 points can be given by

1. $(\overline{r}-\overline{a}).\left(\right(\overline{a}+\overline{b})\times (\overline{b}+\overline{c}\left)\right)=0$
2. $(\overline{r}-\overline{a}).\left(\right(\overline{a}+\overline{b})\times (\overline{b}-\overline{c}\left)\right)=0$
3. $(\overline{r}-\overline{b}).\left(\right(\overline{a}-\overline{b})\times (\overline{b}-\overline{c}\left)\right)=0$
4. $(\overline{r}-\overline{a}).\left(\right(\overline{a}-\overline{b})\times (\overline{b}-\overline{c}\left)\right)=0$

Q. Given 3 points given by position vectors $\overline{a},\overline{b}$ and $\overline{c}$. The plane which passes through these 3 points can be given by

1. 
2. 
3. 
4.