Vectors & 3D Geometry PYQ — Page 10

An arc $PQ$ of a circle subtends a right angle at its centre $O$. The mid point of the arc $PQ$ is $R$. If $\vec{OP}=\vec{u},\vec{OR}=\vec{v}$ and $\vec{OQ}=\alpha \vec{u}+\beta \vec{v}$, then $\alpha ,{\beta }^{2}$, are the roots of the equation

Let $\vec{a}=\hat{i}+2\hat{j}+3\hat{k},\vec{b}=\hat{i}-\hat{j}+2\hat{k}$ and $\vec{c}=5\hat{i}-3\hat{j}+3\hat{k}$, be there(three) vector. If $\vec{r}$ is a vector such that, $\vec{r}\times \vec{b}=\vec{c}\times \vec{b}$ and $\vec{r}\cdot \vec{a}=0$, then $25{|\vec{r}|}^{2}$ is equal to

Let the shortest distance between the lines $L:\frac{x-5}{-2}=\frac{y-\lambda }{0}=\frac{z+\lambda }{1},\lambda \geq 0$ and ${L}_{1}:x+1=y-1=4-z$ be $2\sqrt{6}$. If $(\alpha ,\beta ,\gamma )$ lies on $L$, then which of the following is NOT possible?

The sum of all values of $\alpha$, for which the points whose position vectors are $\hat{i}-2\hat{j}+3\hat{k},2\hat{i}-3\hat{j}+4\hat{k},(\alpha +1)\hat{i}+2\hat{k}$ and $9\hat{i}+(\alpha -8)\hat{j}+6\hat{k}$ are coplanar, is equal to

Let $O$ be the origin and the position vector of the point $P$ be $-\hat{i}-2\hat{j}+3k$. If the position vectors of the points $A,B$ and $C$ are $-2\hat{i}+\hat{j}-3k,2\hat{i}+4\hat{j}-2k$ and $-4\hat{i}^+2\hat{j}-k$ respectively, then the projection of the vector $\vec{OP}$ on a vector perpendicular to the vectors $\vec{AB}$ and $\vec{AC}$ is

Let $\vec{a}=\hat{i}+4\hat{j}+2\hat{k},\vec{b}=3\hat{i}-2\hat{j}+7\hat{k}$ and $\vec{c}=2\hat{i}-\hat{j}+4\hat{k}$. If a vector $\vec{d}$ satisfies $\vec{d}\times \vec{b}=\vec{c}\times \vec{b}$ and $\vec{d}\cdot \vec{a}=24,$ then ${|\vec{d}|}^{2}$ 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 $\vec{\alpha }=4\hat{i}+3\hat{j}+5\hat{k}$ 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

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 shortest distance between the lines $\frac{x-2}{3}=\frac{y+1}{2}=\frac{z-6}{2}$ and $\frac{x-6}{3}=\frac{1-y}{2}=\frac{z+8}{0}$ is equal to ______

If the vectors $\vec{a}=\lambda \hat{i}+\mu \hat{j}+4\hat{k},\vec{b}=-2\hat{i}+4\hat{j}-2\hat{k}$ and $\vec{c}=2\hat{i}+3\hat{j}+\hat{k}$ are coplanar and the projection of $\vec{a}$ on the vector $\vec{b}$ is $\sqrt{54}$ units, then the sum of all possible values of $\lambda +\mu$ is equal to

If the shortest distance between the line joining the points $(1,2,3)$ and $(2,3,4)$, and the line $\frac{x-1}{2}=\frac{y+1}{-1}=\frac{z-2}{0}$ is $\alpha$, then $28{\alpha }^{2}$ is equal to _____ .

Consider the lines ${L}_{1}$ and ${L}_{2}$ given by ${L}_{1}:\frac{x-1}{2}=\frac{y-3}{1}=\frac{z-2}{2}$ ${L}_{2}:\frac{x-2}{1}=\frac{y-2}{2}=\frac{z-3}{3}$ A line ${L}_{3}$ having direction ratios $1,-1,-2$, intersects ${L}_{1}$ and ${L}_{2}$ at the points $P$ and $Q$ respectively. Then the length of line segment $PQ$ 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 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

$A(2,6,2),B(-4,0,\lambda ),C(2,3,-1)$ and $D(4,5,0),|\lambda |\leq 5$ are the vertices of a quadrilateral $ABCD$. If its area is $18$ square units, then $5-6\lambda$ is equal to _____ .

Let ${\lambda }_{1},{\lambda }_{2}$ be the values of $\lambda$ for which the points $(\frac{5}{2},1,\lambda )$ and $(-2,0,1)$ are at equal distance from the plane $2x+3y-6z+7$ If ${\lambda }_{1}>{\lambda }_{2}$ then the distance of the point $({\lambda }_{1}-{\lambda }_{2},{\lambda }_{2},{\lambda }_{1})$ from the line $\frac{x-5}{1}=\frac{y-1}{2}=\frac{z+7}{2}$ is $______$

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

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

Let 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 ________.

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 $a,b,c$ be three distinct real numbers, none equal to one. If the vectors $a\hat{i}+\hat{j}+\hat{k},\hat{i}+b\hat{j}+\hat{k}$ and $\hat{i}+\hat{j}+c\hat{k}$ are coplanar, then $\frac{1}{1-a}+\frac{1}{1-b}+\frac{1}{1-c}$ is equal to