Equation of a Plane Passing through a Point and Perpendicular to a Given Vector

Q. Let $S$ be the reflection of a point $Q$ with respect to the plane given by
$\overrightarrow{r}=-(t+p)\hat{i}+t\hat{j}+(1+p)\hat{k}$
where $t,p$ are real parameters and $\hat{i},\hat{j},\hat{k}$ are the unit vectos along the three positive coordinates axes. If the position vectors of $Q$ and $S$ are $10\hat{i}+15\hat{j}+20\hat{k}$ and $\alpha \hat{i}+\beta \hat{j}+\gamma \hat{k}$ respectively, then which of the following is/are TRUE ?

1. $3(\alpha +\beta +\gamma )=-121$
2. $3(\gamma +\alpha )=-86$
3. $3(\beta +\gamma )=-71$
4. $3(\alpha +\beta )=-101$

Q. If the lengths of the sides of a rectangular parallelopiped are $3,2,1$ then the angle between two diagonals out of four diagonals can be

1. ${\cos }^{-1}\frac{2}{7}$
2. ${\cos }^{-1}\frac{4}{7}$
3. ${\cos }^{-1}\frac{3}{7}$
4. ${\cos }^{-1}\frac{1}{7}$

Q.

Equation of plane passing through a point A given by position vector $\overline{a}$ and perpendicular to $\hat{n}is(\overline{r}-\overline{a}).\hat{n}=0$ .

1. True

2. False

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 X and Y component of p vector are 7i and 6j .Also , the X and Y component of P vector + Q vector are 11i and 9j respectively . Then magnitude of vector Q

Q. Statement $1:$ Scalar components of the vector with initial point $(2,1)$ and terminal point $(-5,7)$ are $-6$ and $-7.$
Statement $2:$ Vector components of the vector with initial point $(2,1)$ and terminal point $(-5,7)$ are $-7\hat{i}$ and $6\hat{j}$.

1. Only statement $1$ is correct
2. Only statement $2$ is correct
3. both statements are true
4. both statements are false

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. Points whose position vectors are $2\hat{i}+\hat{j}-\hat{k},3\hat{i}-2\hat{j}+\hat{k}\text{and}\hat{i}+4\hat{j}-3\hat{k}$ will

1. Form an equilateral triangle
2. Form a right triangle
3. Form a scalene triangle
4. Collinear

Q. If $\overrightarrow{b}$ is the vector whose initial point divides the joining $5\overrightarrow{i}$ and $5\overrightarrow{j}$ in the ratio $\lambda :1$ and terminal point is at origin. If $|\overrightarrow{b}|\le \sqrt{37},$ then set of values of $\lambda$ is

1. $(-\infty ,-6]\cup [-\frac{1}{6},\infty )$
2. $(-6,6)$
3. $(-\infty ,-2)\cup (2,\infty )$
4. $[-2,2]$

Q. Find the vector equation of a plane passing through a point with position vector $2\hat{i}-\hat{j}+\hat{k}$ and perpendicular to the vector $4\hat{i}+2\hat{j}-3\hat{k}.$

Q. If $\overline{a},\overline{b},\overline{c}$ are three vectors such that $\left|\overline{a}\right|=5,\left|\overline{b}\right|=12,\left|\overline{c}\right|=13$ and $\overline{a},\overline{b},\overline{c}$ are perpendicular to $\overline{b}+\overline{c},\overline{c}+\overline{a},\overline{a}+\overline{b}$ respectively, then $|\overline{a}+\overline{b}+\overline{c}|=$

1. $\frac{1}{\sqrt{13}}$
2. $2\sqrt{13}$
3. $\frac{\sqrt{2}}{13}$
4. $13\sqrt{2}$

Q. The equation of the plane $\overrightarrow{r}=\hat{i}-\hat{j}+\lambda \left(\hat{i}+\hat{j}+\hat{k}\right)+\mu \left(\hat{i}-2\hat{j}+3\hat{k}\right)$ in scalar product form is
(a) $\overrightarrow{r}·\left(5\hat{i}-2\hat{j}-3\hat{k}\right)=7$
(b) $\overrightarrow{r}·\left(5\hat{i}+2\hat{j}-3\hat{k}\right)=7$
(c) $\overrightarrow{r}·\left(5\hat{i}-2\hat{j}+3\hat{k}\right)=7$
(d) None of these

Q.

Let and. Find a vector which is perpendicular to both and, and.

Q.

Equation of plane passing through points given by position vectors $\overline{a}and\overline{b}$ and origin will be

1. 

2. 

3. 

4. 

Q. Assertion :Let $\overline{a},\overline{b},\overline{r}$ be the vectors such that $\overline{r}+\overline{r}\times \overline{a}=\overline{b}$ then ${\left|\overline{r}\right|}^2=\frac{{(\overline{a}.\overline{b})}^2+{\left|\overline{b}\right|}^2}{1+{\left|\overline{a}\right|}^2}$. Reason: $\overline{r}=\frac{(\overline{a}.\overline{b})\overline{a}+\overline{b}+\overline{a}\times \overline{b}}{1+{\left|\overline{a}\right|}^2}.$

1. Both Assertion & Reason are individually true & Reason is correct explanation of Assertion
2. Both Assertion & Reason are individually true but Reason is not the correct (proper) explanation of Assertion
3. Assertion is true but Reason is false
4. Assertion is false but Reason is true

Q. The vector equation of a plane which is at a distance of 9 units from the origin and which is normal to the
vector $2\hat{i}-\hat{j}+2\hat{k}$ is

1. $\overrightarrow{r}.(2\hat{i}-\hat{j}+2\hat{k})=27$
2. $\overrightarrow{r.}(2\hat{i}-\hat{j}+2\hat{k})=9$
3. $\overrightarrow{r}.(2\hat{i}-\hat{j}+2\hat{k})=3$

Q. On the Cartesian plane. PQR is an isosceles triangle.
The perimeter of $\Delta PQR$ is

1. 18 units
2. 16 units
3. 20 units
4. 22 units

Q. Find the Cartesian equation of the following planes:
$\overrightarrow{r}\cdot \left[\right(s-2t)\hat{i}+(3-t)\hat{j}+(2s+t\left)\hat{k}\right]=15$.

Q. Let $\overrightarrow{\alpha }=(\lambda -2)\overrightarrow{a}+\overrightarrow{b}$ and $\overrightarrow{\beta }=(4\lambda -2)\overrightarrow{a}+3\overrightarrow{b}$ be two given vectors such that $\overrightarrow{a}$ and $\overrightarrow{b}$ are non-collinear. Then the value of $\lambda$ for which vectors $\overrightarrow{\alpha }$ and $\overrightarrow{\beta }$ are collinear is :

1. $-3$
2. $4$
3. $3$
4. $-4$

Q. Find a unit normal vector to the plane x + 2y + 3z − 6 = 0.

Q. Let $\overrightarrow{a}$ be a vector parallel to the line of intersection of the planes $P_1$ and $P_2.$ Plane $P_1$ is parallel to the vectors $2\hat{j}+3\hat{k}$ and $4\hat{j}-3\hat{k}.$ Plane $P_2$ is parallel to the vectors $\hat{j}-\hat{k}$ and $3\hat{i}+3\hat{j}.$ Then the angle between the vector $\overrightarrow{a}$ and a given vector $2\hat{i}+\hat{j}-2\hat{k}$ can be

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

Q.

Equation of plane passing through a point A given by position vector $\overline{a}$ and perpendicular to $\hat{n}is(\overline{r}-\overline{a}).\hat{n}=0$ .

1. True

2. False

Q. If A=(1, 3, -5) and B=(3, 5, -3), then the vector equation of the plane passing through the midpoint of AB and perpendicular to AB is

1. $r.(\hat{i}+\hat{j}+\hat{k})=2$
2. $r.(\hat{i}+\hat{j}-\hat{k})=2$
3. $r.(\hat{i}-\hat{j}+\hat{k})=2$
   
4. $None$

Q. Indicate the point of the complex plane $z$ which satisfy the following equation.
Im $z^2=0$

Q. The position vectors of points A and B are $\hat{i}-\hat{j}+3\hat{k}$ and $3\hat{i}+3\hat{j}+3\hat{k}$ respectively. The equation of a plane is $\overline{r}(5\hat{i}+2\hat{j}-7\hat{k})+9=0$. The points A and B :

1. are on the opposite side of the plane
2. nothing can be said
3. lie on the plane
4. are on the same side of the plane

Q. The vector equation of a plane which is at a distance of 9 units from the origin and which is normal to the
vector $2\hat{i}-\hat{j}+2\hat{k}$ is

1. $\overrightarrow{r}.(2\hat{i}-\hat{j}+2\hat{k})=27$
2. $\overrightarrow{r.}(2\hat{i}-\hat{j}+2\hat{k})=9$
3. $\overrightarrow{r}.(2\hat{i}-\hat{j}+2\hat{k})=3$

Q.

Find the values of x and y so that the vectors are equal

Q. Find the vector equation of a plane which is at a distance of 3 units from the origin and has $\hat{k}$ as the unit vector normal to it.

Q. Find the vector equation of a plane which is at a distance of 5 units from the origin and which is normal to the vector $\hat{i}-2\hat{j}-2\hat{k}.$

Q. Find the distance of the point $(-1,-5,-10)$ from the point of intersection of the line $\overrightarrow{r}=(2\hat{i}-\hat{j}+2\hat{k})+\lambda (3\hat{i}+4\hat{j}+2\hat{k})$ and the plane $\overrightarrow{r}.(\hat{i}-\hat{j}+\hat{k})=5$