Vectors & 3D Geometry PYQ — Page 9

Let ${L}_{1}:\vec{r}=(\hat{i}-\hat{j}+2\hat{k})+\lambda (\hat{i}-\hat{j}+2\hat{k}),\lambda \in R$, ${L}_{2}:\vec{r}=(\hat{j}-\hat{k})+\mu (3\hat{i}+\hat{j}+p\hat{k}),\mu \in R$ and ${L}_{3}:\vec{r}=\delta (l\hat{i}+m\hat{j}+n\hat{k}),\delta \in R$ be three lines such that ${L}_{1}$ is perpendicular to ${L}_{2}$ and ${L}_{3}$ is perpendicular to both ${L}_{1}$ and ${L}_{2}$. Then the point which lies on ${L}_{3}$ is

Let $\vec{a}=2 \hat{i}+\alpha \hat{j}+\hat{k}, \vec{b}=-\hat{i}+\hat{k}, \vec{c}=\beta \hat{j}-\hat{k}$, where $\alpha$ and $\beta$ are integers and $\alpha \beta=-6$. Let the values of the ordered pair $(\alpha, \beta)$, for which the area of the parallelogram of diagonals $\vec{a}+\vec{b}$ and $\vec{b}+\vec{c}$ is $\frac{\sqrt{21}}{2}$, be $\left(\alpha_1, \beta_1\right)$ and $\left(\alpha_2, \beta_2\right)$. Then $\alpha_1^2+\beta_1^2-\alpha_2 \beta_2$ is equal to

Let $ABCD$ be a quadrilateral. If $E$ and $F$ are the mid points of the diagonals $AC$ and $BD$ respectively and $(\vec{AB}-\vec{BC})+(\vec{AD}-\vec{DC})=k\vec{FE}$, then $k$ is equal to

The shortest distance between the lines $x+1=2y=-12z$ and $x=y+2=6z-6$ is

If the lines $\frac{x-1}{2}=\frac{2-y}{-3}=\frac{z-3}{\alpha }$ and $\frac{x-4}{5}=\frac{y-1}{2}=\frac{z}{\beta }$ intersect, then the magnitude of the minimum value of $8\alpha \beta$ is _____.

Let $\vec{a}=2\hat{i}+\hat{j}+\hat{k}$, and $\vec{b}$ and $\vec{c}$ be two nonzero vectors such that $|\vec{a}+\vec{b}+\vec{c}|=|\vec{a}+\vec{b}-\vec{c}|$ and $\vec{b}\cdot \vec{c}=0$. Consider the following two statement: $(A)|\vec{a}+\lambda \vec{c}|\geq |\vec{a}|$ for all $\lambda \in \mathbb{R}$. $(B)$ $\vec{a}$ and $\vec{c}$ are always parallel

Let the line $L$ pass through the point $(0,1,2),$ intersect the line $\frac{x-1}{2}=\frac{y-2}{3}=\frac{z-3}{4}$ and be parallel to the plane $2x+y-3z=4$. Then the distance of the point $P(1,–9,2)$ from the line $L$ is

Let a line $L$ pass through the point $P(2,3,1)$ and be parallel to the line $x+3y-2z-2=0=x-y+2z$. If the distance of $L$ from the point $(5,3,8)$ is $\alpha$, then $3{\alpha }^{2}$ is equal to ________

Let $\vec{a}=3\hat{i}+\hat{j}-\hat{k}$ and $\vec{c}=2\hat{i}-3\hat{j}+3\hat{k}.$ If $\vec{b}$ is a vector such that $\vec{a}=\vec{b}\times \vec{c}$ and $|\vec{b}{|}^{2}=50,$ then $|72-{|\vec{b}+\vec{c}|}^{2}|$ is equal to $__________.$

The distance of the point $P(4,6,-2)$ from the line passing through the point $(-3,2,3)$ and parallel to a line with direction ratios $3,3,-1$ is equal to:

If the line $x=y=z$ intersects the line $x\mathrm{sin}A+y\mathrm{sin}B+z\mathrm{sin}C-18=0=x\mathrm{sin}2A+y\mathrm{sin}2B+z\mathrm{sin}2C-9$, where $A,B,C$ are the angles of a triangle $ABC$, then $80(\mathrm{sin}\frac{A}{2}\mathrm{sin}\frac{B}{2}\mathrm{sin}\frac{C}{2})$ is equal to _________.

Let $\vec{a}$ and $\vec{b}$ be two vector such that $|\vec{a}|=\sqrt{14},|\vec{b}|=\sqrt{6}$ and $|\vec{a}\times \vec{b}|=\sqrt{48}$. Then ${(\vec{a}\cdot \vec{b})}^{2}$ is equal to _____ .

Let $\vec{a},\vec{b},\vec{c}$ be three vectors such that $|\vec{a}|=\sqrt{31},4|\vec{b}|=|\vec{c}|=2$ and $2(\vec{a}\times \vec{b})=3(\vec{c}\times \vec{a})$. If the angle between $\vec{b}$ and $\vec{c}$ is $\frac{2\pi }{3}$, then ${(\frac{\vec{a}\times \vec{c}}{\vec{a}\cdot \vec{b}})}^{2}$ is equal to _____ .

Let $|\vec{a}|=2,|\vec{b}|=3$ and the angle between the vectors $\vec{a}$ and $\vec{b}$ be $\frac{\pi }{4}$. Then $|(\vec{a}+2\vec{b})\times (2\vec{a}-3\vec{b}){|}^{2}$ is equal to

Let $\vec{a},\vec{b}$ and $\vec{c}$ be three non-zero non-coplanar vectors. Let the position vectors of four points $A,B,C$ and $D$ be $\vec{a}-\vec{b}+\vec{c},\lambda \vec{a}-3\vec{b}+4\vec{c}$, $-\vec{a}+2\vec{b}-3\vec{c}$ and $2\vec{a}-4\vec{b}+6\vec{c}$ respectively. If $\vec{AB}$, $\vec{AC}$ and $\vec{AD}$ are coplanar, then $\lambda$ is :

Let $\vec{a}=2\hat{i}-7\hat{j}+5\hat{k},\vec{b}=\hat{i}+\hat{k}$ and $\vec{c}=\hat{i}+2\hat{j}-3\hat{k}$ be three given vectors. If $\vec{r}$ is a vector such that $\vec{r}\times \vec{a}=\vec{c}\times \vec{a}$ and $\vec{r}\cdot \vec{b}=0$, then $|\vec{r}|$ is equal to:

Let $\vec{a}=2\hat{i}+7\hat{j}-\hat{k},\hat{b}=3\hat{i}+5\hat{k}$ and $\vec{c}=\hat{i}-\hat{j}+2\hat{k}$ Let $\vec{d}$ be a vector which is perpendicular to both $\vec{a}$ and $\vec{b}$, and $\vec{c}\cdot \vec{d}=12$. Then$(-\hat{i}+\hat{j}-\hat{k})\cdot (\vec{c}\times \vec{d})$ is equal to

Let $\vec{a}$ be a non-zero vector parallel to the line of intersection of the two planes described by $\hat{i}+\hat{j},\hat{i}+\hat{k}$ and $\hat{i}-\hat{j},\hat{j}-\hat{k}$. If $\theta$ is the angle between the vector $\vec{a}$ and the vector $\vec{b}=2\hat{i}-2\hat{j}+\hat{k}$ and $\vec{a}\cdot \vec{b}=6$, then the ordered pair $(\theta ,|\vec{a}\times \vec{b}|)$ is equal to

The foot of perpendicular of the point $(2,0,5)$ on the line $\frac{x+1}{2}=\frac{y-1}{5}=\frac{z+1}{-1}$ is $(\alpha ,\beta ,\gamma )$. Then. Which of the following is NOT correct?

Let $N$ be the foot of perpendicular from the point$P(1,-2,3)$ on the line passing through the points $(4,5,8)$ and $(1,-7,5)$. Then the distance of $N$ from the plane $2x-2y+z+5=0$ is

If the shortest distance between the lines $\frac{x+\sqrt{6}}{2}=\frac{y-\sqrt{6}}{3}=\frac{z-\sqrt{6}}{4}$ and $\frac{x-\lambda }{3}=\frac{y-2\sqrt{6}}{4}=\frac{z+2\sqrt{6}}{5}$ is $6$, then sum of squares of all possible values(s) of $\lambda$ is

Let $\vec{a}=4\hat{i}+3\hat{j}$ and $\vec{b}=3\hat{i}-4\hat{j}+5\hat{k}$ and $\vec{c}$ is a vector such that $\vec{c}\cdot (\vec{a}\times \vec{b})+25=0,\vec{c}\cdot (\hat{i}+\hat{j}+\hat{k})=4$ and projection of $\vec{c}$ on $\vec{a}$ is $1$ , then the projection of $\vec{c}$ on $\vec{b}$ equals:

The line ${l}_{1}$ passes through the point $(2,6,2)$ and is perpendicular to the plane $2x+y-2z=10$. Then the shortest distance between the line ${l}_{1}$ and the line $\frac{x+1}{2}=\frac{y+4}{-3}=\frac{z}{2}$ is:

The equation of the plane passing through (1,2,3) and perpendicular to the vector 2i+3j-k is: