Vectors & 3D Geometry PYQ — Page 10

Let $\vec{a}=\hat{i}+2\hat{j}+3\hat{k}$ and $\vec{b}=\hat{i}+\hat{j}-\hat{k}$. If $\vec{c}$ is a vector such that $\vec{a}\cdot \vec{c}=11,\vec{b}\cdot (\vec{a}\times \vec{c})=27$ and $\vec{b}\cdot \vec{c}=$ $-\sqrt{3}|\vec{b}|$, then $|\vec{a}\times \vec{c}{|}^{2}$ 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 area of the quadrilateral $ABCD$ with vertices $A(2,1,1),B(1,2,5),C(-2,-3,5)$ and $D(1,-6,-7)$ is equal to

Let the vectors $\vec{{u}_{1}}=\hat{i}+\hat{j}+a\hat{k},\vec{{u}_{2}}=\hat{i}+b\hat{j}+\hat{k},$ and $\vec{{u}_{3}}=c\hat{i}+\hat{j}+\hat{k}$ be coplanar. If the vectors $\vec{{v}_{1}}=(a+b)\hat{i}+c\hat{j}+c\hat{k},\vec{{v}_{2}}=a\hat{i}+(b+c)\hat{j}+a\hat{k}$ and ${\vec{v}}_{3}=b\hat{i}+b\hat{j}+(c+a)\hat{k}$ are also coplanar, then $6(a+b+c)$ is equal to

If the points with position vectors $\alpha \hat{i}+10\hat{j}+13\hat{k},6\hat{i}+11\hat{j}+11\hat{k},\frac{9}{2}\hat{i}+\beta \hat{j}-8\hat{k}$ are collinear, then ${(19\alpha -6\beta )}^{2}$ is equal to

Let $\vec{a}=6\hat{i}+9\hat{j}+12\hat{k},\vec{b}=\alpha \hat{i}+11\hat{j}-2\hat{k}$ and $\vec{c}$ be vectors such that $\vec{a}\times \vec{c}=\vec{a}\times \vec{b}$ If $\vec{a}\cdot \vec{c}=-12,$ and $\vec{c}\cdot (\hat{i}-2\hat{j}+\hat{k})=5$ then $\vec{c}\cdot (\hat{i}+\hat{j}+\hat{k})$ is equal to $_______$

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:

If $\vec{a}=\hat{i}+2\hat{k},\vec{b}=\hat{i}+\hat{j}+\hat{k},\vec{c}=7\hat{i}-3\hat{j}+4\hat{k},\vec{r}\times \vec{b}+\vec{b}\times \vec{c}=\vec{0}$ and $\vec{r}\cdot \vec{a}=0$ then $\vec{r}.\vec{c}$ is equal to:

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:

Let $\vec{a}$and $\vec{b}$ be two vectors. Let $|\vec{a}|=1,|\vec{b}|=4$ and $\vec{a}\cdot \vec{b}=2$. If $\vec{c}=(2\vec{a}\times \vec{b})-3\vec{b}$, then the value of $\vec{b}\cdot \vec{c}$ is

The vector $\vec{a}=-\hat{i}+2\hat{j}+\hat{k}$ is rotated through a right angle, passing through the $y$-axis in its way and the resulting vector is $\vec{b}$. Then the projection of $3\vec{a}+\sqrt{2}\vec{b}$ on $\vec{c}=5\hat{i}+4\hat{j}+3\hat{k}$ is

Let $\vec{a}=\hat{i}+2\hat{j}+\lambda \hat{k}$, $\vec{b}=3\hat{i}-5\hat{j}-\lambda \hat{k}$, $\vec{a}\cdot \vec{c}=7$, $2(\vec{b}\cdot \vec{c})+43=0$, $\vec{a}\times \vec{c}=\vec{b}\times \vec{c}$, then $\vec{a}\cdot \vec{b}$ is equal to

Let $\vec{a}=-\hat{i}-\hat{j}+\hat{k},\vec{a}\cdot \vec{b}=1$ and $\vec{a}\times \vec{b}=\hat{i}-\hat{j}$. Then $\vec{a}-6\vec{b}$ is equal to

Let $\vec{u}=\hat{i}-\hat{j}-2\hat{k},\vec{v}=2\hat{i}+\hat{j}-\hat{k}$, $\vec{v}\cdot \vec{w}=2$ and $\vec{v}\times \vec{w}=\vec{u}+\lambda \vec{v}$, then $\vec{u}\cdot \vec{w}$ is equal to

Let $\vec{\alpha }=4\hat{i}+3\hat{j}+5\hat{k}$ and $\vec{\beta }=\hat{i}+2\hat{j}-4\hat{k}$. Let ${\vec{\beta }}_{1}$ be parallel to $\vec{\alpha }$ and ${\vec{\beta }}_{2}$ be perpendicular to $\vec{\alpha }$. If $\vec{\beta }={\vec{\beta }}_{1}+{\vec{\beta }}_{2}$, then the value of $5{\vec{\beta }}_{2}\cdot (\hat{i}+\hat{j}+\hat{k})$ is

Let $S$ be the set of all values of $\lambda$, for which the shortest distance between the lines $\frac{x-\lambda }{0}=\frac{y-3}{4}=\frac{z+6}{1}$and $\frac{x+\lambda }{3}=\frac{y}{-4}=\frac{z-6}{0}$ is $13$. Then $8|\underset{\lambda \in S}{\sum }\lambda |$ is equal to

Let a line $L$ pass through the origin and be perpendicular to the lines ${L}_{1}:\vec{r}=(\hat{i}-11\hat{j}-7\hat{k})+\lambda (\hat{i}+2\hat{j}+3\hat{k}),\lambda \in \mathbb{R}$ and ${L}_{2}:\vec{r}=(-\hat{i}+\hat{k})+\mu (2\hat{i}+2\hat{j}+\hat{k}),\mu \in \mathbb{R}$. If $P$ is the point of intersection of $L$ and ${L}_{1}$, and ,Q\alpha ,\beta ,\gamma is the foot of perpendicular from $P$ on ${L}_{2}$, then $9(\alpha +\beta +\gamma )$ 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 _________.

The shortest distance between the lines $\frac{x-4}{4}=\frac{y+2}{5}=\frac{z+3}{3}$ and $\frac{x-1}{3}=\frac{y-3}{4}=\frac{z-4}{2}$ 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 the line $\frac{x}{1}=\frac{6-y}{2}=\frac{z+8}{5}$ intersect the lines $\frac{x-5}{4}=\frac{y-7}{3}=\frac{z+2}{1}$ and $\frac{x+3}{6}=\frac{3-y}{3}=\frac{z-6}{1}$ at the points $A$ and $B$ respectively. Then the distance of the mid-point of the line segment $AB$ from the plane $2x-2y+z=14$ is

Let the image of the point $P(1,2,3)$ in the plane $2x–y+z=9$ be $Q.$ If the coordinates of the point $R$ are $(6,10,7),$ then the square of the area of the triangle $PQR$ is $_______.$

The shortest distance between the lines $\frac{x-5}{1}=\frac{y-2}{2}=\frac{z-4}{-3}$ and $\frac{x+3}{1}=\frac{y+5}{4}=\frac{z-1}{-5}$ is

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: