Vectors & 3D Geometry Questions (38)

If $\vec{a}=\hat{i}+2\hat{j}+3\hat{k}$, $\vec{b}=2\hat{i}+7\hat{j}+2\hat{k}$, $\vec{x}\cdot\vec{a}=0$ and $\vec{x}\cdot\vec{c}=0$ for some non-zero vector $\vec{x}$, then value of $\vec{a}\cdot(\vec{b}\times\vec{c})$ is

A ladder of 3m length leans against a wall. The ladder forms a vertical angle of 30° with the wall. The top slides down at 20 cm/s. The bottom slides away at 20 cm/s at time $t$. The average velocity of a person halfway up the ladder for the first $t$ seconds is

Let $\vec{a}$, $\vec{b}$, $\vec{c}$ be three non-zero vectors satisfying $\vec{a}=\vec{b}\times\vec{c}+2\vec{b}$, where $|\vec{b}|=|\vec{c}|=2$ and $|\vec{a}|\leq4$. The sum of possible values of $|2\vec{a}+\vec{b}+\vec{c}|$ is

Consider the set of eight vectors $V=\{a\hat{i}+b\hat{j}+c\hat{k}: a,b,c\in\{-1,1\}\}$. The number of ways three non-coplanar vectors can be chosen from $V$ equals

A ladder of 3m length leans against a wall. The ladder forms a vertical angle of 30° with the wall. The top slides down at 20 cm/s. The bottom slides away at 20 cm/s at time $t$. The average velocity of a person halfway up the ladder for the first $t$ seconds is

If $\vec{a}=\hat{i}+2\hat{j}+3\hat{k}$, $\vec{b}=2\hat{i}+7\hat{j}+2\hat{k}$, $\vec{x}\cdot\vec{a}=0$ and $\vec{x}\cdot\vec{c}=0$ for some non-zero vector $\vec{x}$, then value of $\vec{a}\cdot(\vec{b}\times\vec{c})$ is

Let $\vec{p},\vec{q}$ and $\vec{r}$ be three unit vectors satisfying $|\vec{p}-\vec{q}|^2+|\vec{q}-\vec{r}|^2+|\vec{r}-\vec{p}|^2=9$. Then $|2\vec{p}+5\vec{q}+5\vec{r}|$ is equal to

Let $\vec{a}$, $\vec{b}$ be two vectors perpendicular to each other with $|\vec{a}|=2$, $|\vec{b}|=3$ and $\vec{c}\times\vec{a}=\vec{b}$. The least value of $|\vec{c}-\vec{a}|$ is

If $\vec{a}=\hat{i}+2\hat{j}+3\hat{k}$, $\vec{b}=2\hat{i}+7\hat{j}+2\hat{k}$, $\vec{x}\cdot\vec{a}=0$ and $\vec{x}\cdot\vec{c}=0$ for some non-zero vector $\vec{x}$, then value of $\vec{a}\cdot(\vec{b}\times\vec{c})$ is

Let $\vec{p},\vec{q}$ and $\vec{r}$ be three unit vectors satisfying $|\vec{p}-\vec{q}|^2+|\vec{q}-\vec{r}|^2+|\vec{r}-\vec{p}|^2=9$. Then $|2\vec{p}+5\vec{q}+5\vec{r}|$ is equal to

Let $\vec{a}$, $\vec{b}$ be two vectors perpendicular to each other with $|\vec{a}|=2$, $|\vec{b}|=3$ and $\vec{c}\times\vec{a}=\vec{b}$. The least value of $|\vec{c}-\vec{a}|$ is

Let $\vec{a}=3\hat i-4\hat j+5\hat k$, $\vec{b}=\hat i+2\hat j-2\hat k$, $\vec{c}=5\hat i-12\hat j-13\hat k$. If $\vec{r}\times\vec{b}=\vec{b}\times(\vec{a}-\vec{c})$ and $\vec{r}\cdot(\vec{a}+\vec{c})=0$, then $|\vec{r}-36\vec{b}|^2$ is

Minimum value of the expression $(a-b)^2 + \left(4\sqrt{1+a^2} + 2\sqrt{2b-b^2}\right)^2$ is

Let $\vec{p},\vec{q}$ and $\vec{r}$ be three unit vectors satisfying $|\vec{p}-\vec{q}|^2+|\vec{q}-\vec{r}|^2+|\vec{r}-\vec{p}|^2=9$. Then $|2\vec{p}+5\vec{q}+5\vec{r}|$ is equal to

Let $\vec{a}$, $\vec{b}$ be two vectors perpendicular to each other with $|\vec{a}|=2$, $|\vec{b}|=3$ and $\vec{c}\times\vec{a}=\vec{b}$. The least value of $|\vec{c}-\vec{a}|$ is

Consider the set of eight vectors $V=\{a\hat{i}+b\hat{j}+c\hat{k}: a,b,c\in\{-1,1\}\}$. The number of ways three non-coplanar vectors can be chosen from $V$ equals

Volume of parallelopiped determined by vectors $\vec{a},\vec{b},\vec{c}$ is 5. Then volume determined by $3(\vec{a}+\vec{b})$, $(\vec{b}+\vec{c})$ and $2(\vec{c}+\vec{a})$ is

A ladder of 3m length leans against a wall. The ladder forms a vertical angle of 30° with the wall. The top slides down at 20 cm/s. The bottom slides away at 20 cm/s at time $t$. The average velocity of a person halfway up the ladder for the first $t$ seconds is

Let $\vec{a}$, $\vec{b}$, $\vec{c}$ have magnitudes 1, 2, 3 respectively, satisfying $|(\vec{a}\times\vec{b})\cdot\vec{c}|=6$. If $\vec{d}$ is a unit vector coplanar with $\vec{b}$ and $\vec{c}$ such that $\vec{b}\cdot\vec{d}=1$, then $|(\vec{a}\times\vec{c})\cdot\vec{d}|^2+|(\vec{a}\times\vec{c})\times\vec{d}|^2$ is

The equation of the plane $P_1$ parallel to line $L: \frac{x-1}{2}=\frac{y-0}{3}=\frac{z-1}{1}$, at distance 6 from the intersection of $L$ with plane $P: 2x+3y+z-18=0$, and having maximum distance from $Q(5,4,-4)$, is $ax+by+cz+d=0$. The value of $\dfrac{bd}{ac}$ is:

The equation of the plane $P_1$ parallel to line $L: \frac{x-1}{2}=\frac{y-0}{3}=\frac{z-1}{1}$, at distance 6 from the intersection of $L$ with plane $P: 2x+3y+z-18=0$, and having maximum distance from $Q(5,4,-4)$, is $ax+by+cz+d=0$. The value of $\dfrac{bd}{ac}$ is:

Let $\gamma\in\mathbb{R}$ such that $L_1: \frac{x+1}{11}=\frac{y+2}{21}=\frac{z+3}{29}$ and $L_2: \frac{x+3}{16}=\frac{y+2}{11}=\frac{z+4}{\gamma}$ intersect at $R_1$. Match P)$\gamma$; Q)unit normal $\hat{n}$; R)$\overrightarrow{OR_1}$; S)$\overrightarrow{OR_1}\cdot\hat{n}$ with List-II: 1)$-\hat{i}-\hat{j}+\hat{k}$; 2)$\frac{3}{2}$; 3)$1$; 4)$\frac{1}{\sqrt{6}}\hat{i}-\frac{2}{\sqrt{6}}\hat{j}+\frac{1}{\sqrt{6}}\hat{k}$; 5)$\frac{2}{3}$

There is a plane $P$ passing through the line $\dfrac{x+1}{2}=\dfrac{y-1}{1}=\dfrac{z+3}{2}$ and parallel to the $x$-axis. Let $\vec{v}$ be a vector parallel to plane $P$ such that the angle between $\vec{v}$ and $\vec{u}=\hat{i}+\hat{j}+\hat{k}$ is least. There is a circle lying on plane $P$ with centre $C(0,0,-5)$ and radius $\sqrt{10}$. A line $L$ on plane $P$ is tangent to the circle at a point $Q(x,y,z)$ with the least possible value of $z$. Then which of the following is/are true?

Let $\vec{a},\vec{b},\vec{c},\vec{d}$ be position vectors of $A,B,C,D$ with $|\vec{a}|=1$, $|\vec{b}|=2$, $|\vec{c}|=3$, $|\vec{d}|=4$, $(\vec{a}\times\vec{b})\cdot(\vec{c}\times\vec{d})=24$, $\vec{a}\cdot\vec{c}=\frac{3}{2}$, $\vec{b}\cdot\vec{c}>0$. Area of $\triangle ABC=\frac{1}{4}(2\alpha+\beta\sqrt{3})$ and area of $\triangle BCD=\frac{1}{2}(m-n\sqrt{3})$ where $\alpha,\beta,m,n\in\mathbb{N}$. Match P)$\alpha$, Q)$\beta$, R)$m$, S)$n$ with 1)1, 2)3, 3)4, 4)9.

There is a plane $P$ passing through the line $\dfrac{x+1}{2}=\dfrac{y-1}{1}=\dfrac{z+3}{2}$ and parallel to the $x$-axis. Let $\vec{v}$ be a vector parallel to plane $P$ such that the angle between $\vec{v}$ and $\vec{u}=\hat{i}+\hat{j}+\hat{k}$ is least. There is a circle lying on plane $P$ with centre $C(0,0,-5)$ and radius $\sqrt{10}$. A line $L$ on plane $P$ is tangent to the circle at a point $Q(x,y,z)$ with the least possible value of $z$. Then which of the following is/are true?

Let $\vec{a},\vec{b},\vec{c},\vec{d}$ be position vectors of $A,B,C,D$ with $|\vec{a}|=1$, $|\vec{b}|=2$, $|\vec{c}|=3$, $|\vec{d}|=4$, $(\vec{a}\times\vec{b})\cdot(\vec{c}\times\vec{d})=24$, $\vec{a}\cdot\vec{c}=\frac{3}{2}$, $\vec{b}\cdot\vec{c}>0$. Area of $\triangle ABC=\frac{1}{4}(2\alpha+\beta\sqrt{3})$ and area of $\triangle BCD=\frac{1}{2}(m-n\sqrt{3})$ where $\alpha,\beta,m,n\in\mathbb{N}$. Match P)$\alpha$, Q)$\beta$, R)$m$, S)$n$ with 1)1, 2)3, 3)4, 4)9.

Let $\gamma\in\mathbb{R}$ such that $L_1: \frac{x+1}{11}=\frac{y+2}{21}=\frac{z+3}{29}$ and $L_2: \frac{x+3}{16}=\frac{y+2}{11}=\frac{z+4}{\gamma}$ intersect at $R_1$. Match P)$\gamma$; Q)unit normal $\hat{n}$; R)$\overrightarrow{OR_1}$; S)$\overrightarrow{OR_1}\cdot\hat{n}$ with List-II: 1)$-\hat{i}-\hat{j}+\hat{k}$; 2)$\frac{3}{2}$; 3)$1$; 4)$\frac{1}{\sqrt{6}}\hat{i}-\frac{2}{\sqrt{6}}\hat{j}+\frac{1}{\sqrt{6}}\hat{k}$; 5)$\frac{2}{3}$

Let $\vec{a}$, $\vec{b}$, $\vec{c}$ be three non-zero vectors satisfying $\vec{a}=\vec{b}\times\vec{c}+2\vec{b}$, where $|\vec{b}|=|\vec{c}|=2$ and $|\vec{a}|\leq4$. The sum of possible values of $|2\vec{a}+\vec{b}+\vec{c}|$ is

Let $\vec{a}=3\hat i-4\hat j+5\hat k$, $\vec{b}=\hat i+2\hat j-2\hat k$, $\vec{c}=5\hat i-12\hat j-13\hat k$. If $\vec{r}\times\vec{b}=\vec{b}\times(\vec{a}-\vec{c})$ and $\vec{r}\cdot(\vec{a}+\vec{c})=0$, then $|\vec{r}-36\vec{b}|^2$ is

Consider the set of eight vectors $V=\{a\hat{i}+b\hat{j}+c\hat{k}: a,b,c\in\{-1,1\}\}$. The number of ways three non-coplanar vectors can be chosen from $V$ equals

Volume of parallelopiped determined by vectors $\vec{a},\vec{b},\vec{c}$ is 5. Then volume determined by $3(\vec{a}+\vec{b})$, $(\vec{b}+\vec{c})$ and $2(\vec{c}+\vec{a})$ is

Let $\vec{a}$, $\vec{b}$, $\vec{c}$ have magnitudes 1, 2, 3 respectively, satisfying $|(\vec{a}\times\vec{b})\cdot\vec{c}|=6$. If $\vec{d}$ is a unit vector coplanar with $\vec{b}$ and $\vec{c}$ such that $\vec{b}\cdot\vec{d}=1$, then $|(\vec{a}\times\vec{c})\cdot\vec{d}|^2+|(\vec{a}\times\vec{c})\times\vec{d}|^2$ is

Let $\vec{a}$, $\vec{b}$, $\vec{c}$ be three non-zero vectors satisfying $\vec{a}=\vec{b}\times\vec{c}+2\vec{b}$, where $|\vec{b}|=|\vec{c}|=2$ and $|\vec{a}|\leq4$. The sum of possible values of $|2\vec{a}+\vec{b}+\vec{c}|$ is

Minimum value of the expression $(a-b)^2 + \left(4\sqrt{1+a^2} + 2\sqrt{2b-b^2}\right)^2$ is

Let $\vec{a}=3\hat i-4\hat j+5\hat k$, $\vec{b}=\hat i+2\hat j-2\hat k$, $\vec{c}=5\hat i-12\hat j-13\hat k$. If $\vec{r}\times\vec{b}=\vec{b}\times(\vec{a}-\vec{c})$ and $\vec{r}\cdot(\vec{a}+\vec{c})=0$, then $|\vec{r}-36\vec{b}|^2$ is

Minimum value of the expression $(a-b)^2 + \left(4\sqrt{1+a^2} + 2\sqrt{2b-b^2}\right)^2$ is

Let $\vec{a}$, $\vec{b}$, $\vec{c}$ have magnitudes 1, 2, 3 respectively, satisfying $|(\vec{a}\times\vec{b})\cdot\vec{c}|=6$. If $\vec{d}$ is a unit vector coplanar with $\vec{b}$ and $\vec{c}$ such that $\vec{b}\cdot\vec{d}=1$, then $|(\vec{a}\times\vec{c})\cdot\vec{d}|^2+|(\vec{a}\times\vec{c})\times\vec{d}|^2$ is

Volume of parallelopiped determined by vectors $\vec{a},\vec{b},\vec{c}$ is 5. Then volume determined by $3(\vec{a}+\vec{b})$, $(\vec{b}+\vec{c})$ and $2(\vec{c}+\vec{a})$ is