Equation of a Plane Passing through Three Points

Q.

A plane passes through the points $A(1,2,3),B(2,3,1)$ and $C(2,4,2)$. If $O$ is the origin and $P$ is $(2,–1,1)$, then the projection of vector$\left(OP\right)$ on this plane is of length:

1. $\sqrt{\frac{2}{5}}$

2. $\sqrt{\frac{2}{3}}$

3. $\sqrt{\frac{2}{11}}$

4. $\sqrt{\frac{2}{7}}$

Q.

If a plane cuts off intercepts -6, 3, 4 from the co-ordinate axes, then the length of the perpendicular from the origin to the plane is

1. 
   

2. 
   

3. 
   

4. 
   

Q. The direction ratio of the normal to the plane passing through the points $(1,2,-3),$ $(-1,-2,1)$ and parallel to $\frac{x-2}{2}=\frac{y+1}{3}=\frac{z}{4}$ is

1. $(2,3,4)$
2. $(14,-8,-1)$
3. $(-2,0,-3)$
4. $(1,-2,-3)$

Q.

Let P be the image of the point (3, 1, 7) with respect to the plane x - y + z = 3. Then, the equation of the plane passing through P and containing the straight line $\frac{x}{1}=\frac{y}{2}=\frac{z}{1}$ is

1. x + y - 3z = 0

2. 3x + z = 0

3. x - 4y + 7z = 0

4. 2x - y = 0

Q.

Let $P$ be a plane passing through the points $(2,1,0),(4,1,1)$and $(5,0,1)$ and $R$ be any point$(2,1,6).$ Then the image of $R$ in the plane $P$ is:

1. $(6,5,2)$

2. $(6,5,-2)$

3. $(4,3,2)$

4. $(3,4,-2)$

Q. Find the vector and Cartesian equations of the plane passing through the points $(2,2,-1),(3,4,2)\text{and}(7,0,6).$ Also find the vector equation of a plane passing through $(4,3,1)$ and parallel to the plane obtained above.

Q.

The distance of the point $\left(3,8,2\right)$ from the line $\frac{x-1}{2}=\frac{y-3}{4}=\frac{z-2}{3}$ measured parallel to the plane $3x+2y-2z=0$ is

1. $2$

2. $3$

3. $6$

4. $7$

Q.

Find the equation of plane passing through the points P (1, 1, 1) , Q (3, -1, 2) and R (-3, 5 , -4).

1. 6x + 16y +16z - 44 = 0

2. x - 2y +16z - 13 = 0

3. 6x + 6y +16z - 28 = 0

4. x + 6y +6z - 22 = 0

Q. A plane meets the co-ordinates axes in A, B, C and $(\alpha ,\beta ,\gamma )$ is the centroid of the $\Delta ABC$. Then the equation of the plane is :

1. $\frac{3x}{\alpha }+\frac{3y}{\beta }+\frac{3z}{\gamma }=1$
2. $\alpha x+\beta y+\gamma z=1$
3. $\frac{x}{\alpha }+\frac{y}{\beta }+\frac{z}{\gamma }=1$
4. $\frac{x}{\alpha }+\frac{y}{\beta }+\frac{z}{\gamma }=3$

Q. If $\overrightarrow{a}+\overrightarrow{b}+\overrightarrow{c}=\alpha \overrightarrow{d},$ $\overrightarrow{b}+\overrightarrow{c}+\overrightarrow{d}=\beta \overrightarrow{a}$ and $\overrightarrow{a},\overrightarrow{b},\overrightarrow{c}$ are non-coplanar, then the sum of $\overrightarrow{a}+\overrightarrow{b}+\overrightarrow{c}+\overrightarrow{d}=$

1. $\overrightarrow{0}$
2. $(\beta -1)\cdot \overrightarrow{d}+(\alpha -1)\cdot \overrightarrow{a}$
3. $(\alpha -1)\cdot \overrightarrow{d}-(\beta -1)\cdot \overrightarrow{a}$
4. $(\alpha -1)\cdot \overrightarrow{d}+(\beta -1)\cdot \overrightarrow{a}$

Q.

If ${\mathrm{z}}_1=1+\mathrm{i},{\mathrm{z}}_2=–2+3\mathrm{i}$ and ${\mathrm{z}}_3=\frac{{\mathrm{a}}_{\mathrm{i}}}{3}$ where ${\mathrm{i}}^2=–1$, are collinear then the value of $\mathrm{a}$ is

1. $-1$

2. $3$

3. $4$

4. $5$

Q.

Find the vector and Cartesian equations of the planes.

that passes through the point (1, 0, -2) and the normal to the plane is $\hat{i}+\hat{j}-\hat{k}.$

that passes through the point (1, 4, 6) and the normal to the plane is $\hat{i}-2\hat{j}+\hat{k}.$

Q. A variable plane passes through a fixed point $(3,2,1)$ and meets $x,y$ and $z$ axes at $A,B$ and $C$ respectively. A plane is drawn parallel to $yz-$plane through $A$, a second plane is drawn parallel $zx-$plane through $B$ and a third plane is drawn parallel to $xy-$plane through $C$. Then the locus of the point of intersection of these three planes, is :

1. $x+y+z=6$
2. $\frac{x}{3}+\frac{y}{2}+\frac{z}{1}=1$
3. $\frac{3}{x}+\frac{2}{y}+\frac{1}{z}=1$
4. $\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=\frac{11}{6}$

Q.

If four points P (1, 2, 3) , Q (2, 3, 4), R(3, 4, 5), S(4, 5, k) are coplanar then the value of k is / are -

1. 6

2. 4

3. 5

4. Any real number.

Q. A plane passes through the points $A(1,2,3),B(2,3,1)$ and $C(2,4,2).$ If $O$ is the origin and $P$ is $(2,-1,1),$ then the projection of $\overrightarrow{OP}$ on this plane is of length :

1. $\sqrt{\frac{2}{3}}$
2. $\sqrt{\frac{2}{5}}$
3. $\sqrt{\frac{2}{11}}$
4. $\sqrt{\frac{2}{7}}$

Q.

Find the vector and Cartesian equations of the plane,

that passes through the point (1, 4, 6) and the normal to the plane is $\hat{i}-2\hat{j}+\hat{k}.$

Q. Let $P$ be the image of the point $(3,1,7)$ with respect to the plane $x-y+z=3.$ Then the equation of the plane passing through $P$ and containing the straight line $\frac{x}{1}=\frac{y}{2}=\frac{z}{1}$ is

1. $x+y-3z=0$
2. $x-4y+7z=0$
3. $2x-y=0$
4. $3x+z=0$

Q. A perpendicular is drawn from the point $A(1,8,4)$ to the line joining the points $B(0,-1,3)$ and $C(2,-3,-1)$. The co-ordinates of the foot of the perpendicular is:

1. $(\frac{5}{3},\frac{2}{3},\frac{19}{3})$
2. $(\frac{-5}{3},\frac{2}{3},\frac{19}{3})$
3. $(\frac{5}{3},\frac{-2}{3},\frac{19}{3})$
4. $(\frac{5}{3},\frac{2}{3},\frac{-19}{3})$

Q. Find the equation of the plane passing through the points $(1,2,3),(3,5,7)$ and $(4,3,1)$

Q. Show that the line through points (4, 7, 8) and (2, 3, 4) is parallel to the line through the points (−1, −2, 1) and (1, 2, 5).

Q. A variable plane intersects the coordinate axes at $A,B,C$ and is at a constant distance $p$ from $O(0,0,0).$ Then the locus of the centroid of the tetrahedron $OABC$ is

1. $\frac{1}{x^2}+\frac{1}{y^2}+\frac{1}{z^2}=16p^2$
2. $\frac{1}{x^2}+\frac{1}{y^2}+\frac{1}{z^2}=\frac{16}{p^2}$
3. $\frac{1}{x^2}+\frac{1}{y^2}+\frac{1}{z^2}=\frac{1}{p^2}$
4. $\frac{1}{x^2}+\frac{1}{y^2}+\frac{1}{z^2}=\frac{4}{p^2}$

Q.

The position vectors of $P$ and $Q$ are respectively, $a$ and $b$. If $R$ is point on $PQ$ such that $PR=5PQ$, then the position vector $R$ is?

1. $5b–4a$

2. $5b+4a$

3. $4a-5b$

4. $4a+5b$

Q. Equation of the plane through three points $A,B,C$ with position vectors $-6\overrightarrow{i}+3\overrightarrow{j}+2\overrightarrow{k},3\overrightarrow{i}-2\overrightarrow{j}+4\overrightarrow{k},5\overrightarrow{i}+7\overrightarrow{j}+3\overrightarrow{k}$ is

1. $\overrightarrow{r}\cdot (\overrightarrow{i}-\overrightarrow{j}-7\overrightarrow{k})+23=0$
2. $\overrightarrow{r}\cdot (\overrightarrow{i}+\overrightarrow{j}+7\overrightarrow{k})=23$
3. $\overrightarrow{r}\cdot (\overrightarrow{i}+\overrightarrow{j}-7\overrightarrow{k})+23=0$
4. $\overrightarrow{r}\cdot (\overrightarrow{i}-\overrightarrow{j}-7\overrightarrow{k})=23$

Q. A plane passes through the points $A(1,2,3),B(2,3,1)$ and $C(2,4,2).$ If $O$ is the origin and $P$ is $(2,-1,1),$ then the projection of $\overrightarrow{OP}$ on this plane is of length :

1. $\sqrt{\frac{2}{5}}$
2. $\sqrt{\frac{2}{3}}$
3. $\sqrt{\frac{2}{11}}$
4. $\sqrt{\frac{2}{7}}$

Q. A plane moves such that its distance from the origin is a constant p. If it intersects the coordinate axes at A, B,
C then the locus of the centroid of the triangle ABC is

1. $\frac{1}{x^2}+\frac{1}{y^2}+\frac{1}{z^2}=\frac{1}{p^2}$
2. $\frac{1}{x^2}+\frac{1}{y^2}+\frac{1}{z^2}=\frac{9}{p^2}$
3. $\frac{1}{x^2}+\frac{1}{y^2}+\frac{1}{z^2}=\frac{2}{p^2}$
4. $\frac{1}{x^2}+\frac{1}{y^2}+\frac{1}{z^2}=\frac{4}{p^2}$

Q.

If four points P (1, 2, 3) , Q (2, 3, 4), R(3, 4, 5), S(4, 5, k) are coplanar then the value of k is / are -

1. 6

2. 4

3. 5

4. Any real number.

Q. Consider three vectors $\overrightarrow{p}=i+j+k,\overrightarrow{q}=2i+4j-k$ and $\overrightarrow{r}=i+j+3k$. If $p,q$ and $r$ denotes the position vector of three non-collinear points, then the equation of the plane containing these points is

1. $2x-3y+1=0$
2. $x-3y+2z=0$
3. $3x-y+z-3=0$
4. $3x-y-2=0$

Q. The perpendicular distance from the point $(3,1,1)$ on the plane passing through the point $(1,2,3)$ and containing the line, $\overrightarrow{r}=\hat{i}+\hat{j}+\lambda (2\hat{i}+\hat{j}+4\hat{k})$, is:

1. $\frac{1}{\sqrt{11}}$
2. $\frac{4}{\sqrt{41}}$
3. $0$
4. $\frac{3}{\sqrt{11}}$

Q. Vector equation of the plane $\overrightarrow{r}=\widehat{i-}\hat{j}+\lambda (\hat{i}-\hat{j}+\hat{k})+\mu (\hat{i}-2\hat{j}+3\hat{k})$

1. $-\hat{i}-2\hat{j}-\hat{k}$
2. $\overrightarrow{r}.(5\hat{i}+2\hat{j}-3\hat{k})=7$
3. $\overrightarrow{r}.(5\hat{i}-2\hat{j}-3\hat{k})=7$
4. $\overrightarrow{r}.(5\hat{i}+2\hat{j}+3\hat{k})=7$

Q.

Find the Cartesian equation of the following plane:
$r.(2\hat{i}+3\hat{j}-4\hat{k})=1$