Equation of a Plane Passing through a Point and Parallel to the Two Given Vectors

Q. 5. Given vector A = 2i + 3j and vector B = i + j , find the component of vector A along B.

Q. The equation of the plane parallel to the lines $\overrightarrow{r}=\hat{i}+\hat{j}+\hat{k}+\lambda (2\hat{i}+\hat{j}+4\hat{k})$ and $\frac{x+1}{-3}=\frac{y-3}{2}=\frac{z+2}{1}$ and is passing through the point $(0,1,-1)$

1. $x+2y-z=3$
2. $3x-2y-z=6$
3. $3x+2y+z=6$
4. $x-2y-z=5$

Q. The equation of plane passing through the point $(1,1,1)$ and perpendicular to the planes $2x+y-2z=5$ and $3x-6y-2z=7$ is

1. $14x+2y-15z=31$
2. $14x-2y+15z=31$
3. $14x+2y+15z=31$
4. $14x-2y-15z=31$

Q. Let $L_1:\frac{x-1}{3}=\frac{y-2}{1}=\frac{z-3}{-3}$ be a line and $P:4x+3y+5z-50=0$ be a plane. $L_2$ is the line in the plane $P$ and parallel to $L_1.$ If equation of the plane containing both the lines $L_1$ and $L_2$ and perpendicular to plane $P$ is $ax+by+5z+d=0,$ then the value of $a+b+d$ is

Q. Let $L_1:\frac{x-1}{3}=\frac{y-2}{1}=\frac{z-3}{-3}$ be a line and $P:4x+3y+5z=50$ be a plane. $L_2$ is the line in the plane $P$ and parallel to $L_1$. If equation of the plane containing both the lines $L_1$ and $L_2$ and perpendicular to plane $P$ is $ax+by+5z+d=0,$ then the value of $(a+b+d)$ is

Q.

Two lines whose direction ratios are a1, b1, c1 and a2, b2, c2 are parallel, if

1. 

2. 

3. 

4. 

Q. Let a plane $P_1$ containing the line $L_1:\frac{x-1}{1}=\frac{y-1}{-1}=\frac{z-1}{-1}$ and another plane $P_2$ containing the line $L_1$ and passses through the point $(0,1,1).$ If d.r's of normal to the plane $P_1$ are $1,1,0,$ then the acute angle between the planes $P_1$ and $P_2$ is equal to

1. $\frac{\pi }{3}$
2. $\frac{\pi }{6}$
3. $\frac{\pi }{4}$
4. $\frac{\pi }{8}$

Q. The equation of the plane containing the line $\frac{x-x_1}{l}=\frac{y-y_1}{m}=\frac{z-z_1}{n}$ is $a(x-x_1)+b(y-y_1)+c(z-z_1)=0,$ then which of the following is true?

1. $a{x}_1+b{y}_1+c{z}_1=0$
2. $al+bm+cn=0$
3. $\frac{a}{l}=\frac{b}{m}=\frac{c}{n}$
4. $l{x}_1+m{y}_1+n{z}_1=0$

Q. If $ax+by+cz-1=0$ is the plane passing through $(4,-1,2)$ and parallel to the lines $\frac{x+2}{3}=\frac{y-2}{-1}=\frac{z+1}{2}$ and $\frac{x-2}{1}=\frac{y-3}{2}=\frac{z-4}{3},$ then $4a+3b+2c=$

Q. The equation of the plane passing through $(3,4,-1)$ and parallel to the plane $2x-3y+5z-7=0$ is

1. $2x-3y+5z+15=0$
2. $2x-3y+5z+9=0$
3. $2x-3y+5z+11=0$
4. $2x-3y-5z-13=0$

Q. The equation of the plane through the line of intersection of $2x-y+3z+1=0,x+y+z+3=0$ and parallel to the line $\frac{x}{1}=\frac{y}{2}=\frac{z}{3}$ is:

1. $x-5y+3z=7$
2. $x-5y+3z=17$
3. $x+5y+3z=17$
4. $x+5y+3z=-7$

Q. The foot of perpendicular of the point $M(1,-2,1)$ on the plane $P:2x-y+2z+3=0$ is

1. $(-3,-1,-3)$
2. $(-2,2,-2)$
3. $(-1,-1,-1)$
4. $(-1,2,-3)$

Q. Consider the following plane equations,
$P_1:-2x-y+z=5P_2:3x-2y+4z=6P_3:4x+2y-2z=6.$
Then which of the following statement(s) is/are correct ?

1. $P_1$ and $P_2$ are parallel.
2. $P_1$ and $P_2$ are perpendicular.
3. $P_1$ and $P_3$ are parallel.
4. $P_2$ and $P_3$ are perpendicular

Q. If the line $\frac{x+1}{-1}=\frac{y-1}{2}=\frac{z-2}{1}$ lies on the plane $nx+my-2z=4,$ then $m-n=$

Q. Let a plane $P_1$ containing the line $L_1:\frac{x-1}{1}=\frac{y-1}{-1}=\frac{z-1}{-1}$ and $P_2$ is another plane containing the line $L_1$ and passes through point $(0,1,1),$ if direction ratio of normal to the plane $P_1$ is $(1,1,0).$ then which of the following statement(s) is/are correct ?

1. The equation of plane $P_1$ is : $x+y=2$
2. The equation of plane $P_1$ is : $x-y=2$
3. The equation of plane $P_2$ is: $x-z=0$
4. The equation of plane $P_2$ is: $y-z=0$

Q. Consider the following equation of planes ,
$P_1:2x-4y+6z=5P_2:-x+2y-3z=8P_3:3x+6y+3z=24.$
Then which of the following is/are correct

1. $P_1$ and $P_2$ are parallel.
2. $P_1$ and $P_3$ are parallel.
3. $P_2$ and $P_3$ are perpendicular.
4. $P_1,P_2$ and $P_3$ are parallel.

Q. For given vector equation of the plane $\overrightarrow{r}=\hat{i}-\hat{j}+\lambda (\hat{i}-\hat{j}+\hat{k})+\mu (\hat{i}-2\hat{j}+3\hat{k}),$ the cartesian equation of the plane is

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

Q. Consider the two plane $P_1:2x-y+2z+3=0,P_2:3x-2y+6z+8=0,$ then which of the following statement(s) is/are correct for angle bisector between plane $P_1$ and $P_2$

1. $5x-y-4z-3=0$ is acute angle bisector
2. $5x-y-4z-3=0$ is obtuse angle bisector
3. $23x-13y+32z+45=0$ is acute angle bisector
4. $23x-13y+32z+45=0$ is obtuse angle bisector

Q.

The equation of the plane containing the lines

$\overrightarrow{r}={\overrightarrow{a}}_1+\lambda {\overrightarrow{a}}_{2}and\overrightarrow{r}={\overrightarrow{a}}_2+\mu {\overrightarrow{a}}_1$ is

1. None of these

2. $\left[\overrightarrow{r}{\overrightarrow{a}}_1{\overrightarrow{a}}_2\right]=0$

3. $\left[\overrightarrow{r}{\overrightarrow{a}}_1{\overrightarrow{a}}_2\right]={\overrightarrow{a}}_1.{\overrightarrow{a}}_2$

4. $\left[\overrightarrow{r}{\overrightarrow{a}}_1{\overrightarrow{a}}_2\right]=|{\overrightarrow{a}}_1||{\overrightarrow{a}}_2|$

Q. The equation of the plane which passes through the line $a_{1}x+b_{1}y+c_{1}z+d_1=0,a_{2}x+b_{2}y+c_{2}z+d_2=0$ and which is parallel to the line $\frac{x-\alpha }{l}=\frac{y-\beta }{m}=\frac{z-\gamma }{n}$ is:

1. $(a_{1}x+b_{1}y+c_{1}z+d_1)(a_{2}l+b_{2}m+c_{2}n)-(a_{1}l+b_{1}m+c_{1}n)(a_{2}x+b_{2}y+c_{2}z+d_2)=0$
2. $(a_{1}x+b_{1}y+c_{1}z+d_1)(a_{2}l+b_{2}m+c_{2}n)+(a_{1}l+b_{1}m+c_{1}n)(a_{2}x+b_{2}y+c_{2}z+d_2)=0$
3. $(a_{1}x+b_{1}y+c_{1}z-d_1)(a_{2}l+b_{2}m+c_{2}n)-(a_{1}l+b_{1}m+c_{1}n)(a_{2}x+b_{2}y+c_{2}z-d_2)=0$
4. $(a_{1}x-b_{1}y-c_{1}z+d_1)(a_{2}l+b_{2}m+c_{2}n)-(a_{1}l+b_{1}m+c_{1}n)(a_{2}x-b_{2}y-c_{2}z+d_2)=0$

Q. Number of planes possible satisfying the condition that it should be parallel to 2 given vectors.

1. 
2. 
3. 
4. 

Q.

Equation of a plane passing through $\hat{i}+\hat{j}+\hat{k}$ parallel to both $\hat{i}+\hat{j}and\hat{j}+\hat{k}$ can be given by

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

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

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

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

Q.

If $A_1,A_2,G_1,G_2$ and $H_1,H_2$ are two AMs, GMs and HMs between two quantities, then the value of $\frac{G_1{G}_2}{H_1{H}_2}$?

1. $\frac{(A_1+A_2)}{(H_1+H_2)}$

2. $\frac{(A_1-A_2)}{(H_1+H_2)}$

3. $\frac{(A_1+A_2)}{(H_1-H_2)}$

4. $\frac{(A_1-A_2)}{(H_1-H_2)}$

Q. Number of planes possible satisfying the condition that it should be parallel to 2 given vectors.

1. 
2. 
3. 
4. 

Q.

The equation of the plane containing the lines

$\overrightarrow{r}={\overrightarrow{a}}_1+\lambda {\overrightarrow{a}}_{2}and\overrightarrow{r}={\overrightarrow{a}}_2+\mu {\overrightarrow{a}}_1$ is

1. $\left[\overrightarrow{r}{\overrightarrow{a}}_1{\overrightarrow{a}}_2\right]=0$

2. $\left[\overrightarrow{r}{\overrightarrow{a}}_1{\overrightarrow{a}}_2\right]={\overrightarrow{a}}_1.{\overrightarrow{a}}_2$

3. $\left[\overrightarrow{r}{\overrightarrow{a}}_1{\overrightarrow{a}}_2\right]=|{\overrightarrow{a}}_1||{\overrightarrow{a}}_2|$

4. None of these

Q. Let the line $\frac{x-2}{3}=\frac{y-1}{-5}=\frac{z+2}{2}$ lies in the plane $x+3y-\alpha z+\beta =0.$ Then $(\alpha ,\beta )$ equals

1. $(6,-17)$
2. $(-6,7)$
3. $(5,-15)$
4. $(-5,15)$

Q. Let $L_1:\frac{x-1}{3}=\frac{y-2}{1}=\frac{z-3}{-3}$ be a line and $P:4x+3y+5z-50=0$ be a plane. $L_2$ is the line in the plane $P$ and parallel to $L_1.$ If equation of the plane containing both the lines $L_1$ and $L_2$ and perpendicular to plane $P$ is $ax+by+5z+d=0,$ then the value of $a+b+d$ is

Q. A firm is engaged in producing two models $X_{1.}{X}_2$ performing only three operations assembling, pianting and testing.The relevant data are as follows.
Total number of hours available each week for assemblling $600$ hours, painting $100$ hours and testing $30$ hours.The firm wishes to determine its weekly products mix so as to maximize revenue formulate I.P.P model for maximize the revenue

Q. Number of planes possible satisfying the condition that it should be parallel to 2 given vectors.

1. $0$
2. $\infty$
3. $1$
4. $2$

Q.

Equation of a plane passing through $\hat{i}+\hat{j}+\hat{k}$ parallel to both $\hat{i}+\hat{j}and\hat{j}+\hat{k}$ can be given by

1. 

2. 

3. 

4.