Vectors & 3D Geometry PYQ — Page 9

Consider the line $L$ passing through the points $(1,2,3)$ and $(2,3,5)$. The distance of the point $\left(\frac{11}{3}, \frac{11}{3}, \frac{19}{3}\right)$ from the line $\mathrm{L}$ along the line $\frac{3 x-11}{2}=\frac{3 y-11}{1}=\frac{3 z-19}{2}$ is equal to

If the mirror image of the point $P(3,4,9)$ in the line $\frac{x-1}{3}=\frac{y+1}{2}=\frac{z-2}{1}$ is $(\alpha ,\beta ,\gamma ),$ then $14(\alpha +\beta +\gamma )$ is:

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 $\vec{a}=2\hat{i}+7\hat{j}-\hat{k},\hat{b}=3\hat{i}+5\hat{k}$ 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

$A(2,6,2),B(-4,0,\lambda ),C(2,3,-1)$ 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 scalar triple product [i j k] is:

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

One vertex of a rectangular parallelopiped is at the origin $O$ and the lengths of its edges along $x,y$ and $z$ axes are $3,4$ and $5$ units respectively. Let $P$ be the vertex $(3,4,5).$ Then the shortest distance between the diagonal $OP$ and an edge parallel to $z$ axis, not passing through $O$ or $P$ is

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 $\vec{a}=5\hat{i}-\hat{j}-3\hat{k}$ and $\vec{b}=\hat{i}+3\hat{j}+5\hat{k}$ be two vectors. Then which one of the following statements is TRUE?

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

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

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

Let $\vec{a}=2\hat{i}+3\hat{j}+4\hat{k},\vec{b}=\hat{i}-2\hat{j}-2\hat{k}$ and $\vec{c}=-\hat{i}+4\hat{j}+3\hat{k}.$ If $\vec{d}$ is a vector perpendicular to both $\vec{b}$ and $\vec{c},$ and $\vec{a}\cdot \vec{d}=18,$ then $|\vec{a}\times \vec{d}{|}^{2}$ 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

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 $|\vec{a}|=2,|\vec{b}|=3$ 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 $ABCD$ 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

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 $\vec{a}=3\hat{i}+\hat{j}-\hat{k}$ 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 $__________.$

Let $\lambda \in \mathbb{Z},\vec{a}=\lambda \hat{i}+\hat{j}-\hat{k}$ and $\vec{b}=3\hat{i}-\hat{j}+2\hat{k}$. Let $\vec{c}$ be a vector such that $(\vec{a}+\vec{b}+\vec{c})\times \vec{c}=\vec{0},\vec{a}\cdot \vec{c}=-17$ and $\vec{b}\cdot \vec{c}=-20$. Then ${|\vec{c}\times (\lambda \hat{i}+\hat{j}+\hat{k})|}^{2}$ is equal to

For any vector $\vec{a}={a}_{1}\hat{i}+{a}_{2}\hat{j}+{a}_{3}\hat{k}$, with $10|{a}_{i}|<1,i=1,2,3$, consider the following statements: $(A)$ : $\mathrm{max}{|{a}_{1}|,|{a}_{2}|,|{a}_{3}|}\leq |\vec{a}|$ $(B)$ : $|\vec{a}|\leq 3\mathrm{max}{|{a}_{1}|,|{a}_{2}|,|{a}_{3}|}$