Vectors & 3D Geometry PYQ — Page 11

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 $\vec{a}$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

If the lines $\frac{x-1}{1}=\frac{y-2}{2}=\frac{z+3}{1}$ and $\frac{x-a}{2}=\frac{y+2}{3}=\frac{z-3}{1}$ intersects at the point $P$, then the distance of the point $P$ from the plane $z=a$ 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 $\vec{a}=-\hat{i}-\hat{j}+\hat{k},\vec{a}\cdot \vec{b}=1$ and $\vec{a}\times \vec{b}=\hat{i}-\hat{j}$. Then $\vec{a}-6\vec{b}$ is equal to

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

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 $\vec{a}=2\hat{i}+3\hat{j}+4\hat{k},\vec{b}=\hat{i}-2\hat{j}-2\hat{k}$ 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

Let $S$ be the set of all $(\lambda ,\mu )$ for which the vectors $\lambda \hat{i}-\hat{j}+\hat{k},\hat{j}+2\hat{j}+\mu \hat{k}$ and $3\hat{i}-4\hat{j}+5\hat{k}$, where $\lambda -\mu =5$, are coplanar, then $\underset{(\lambda ,\mu )\in S}{\sum }80({\lambda }^{2}+{\mu }^{2})$ is equal to

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 $\frac{x+2}{1}=\frac{y}{-2}=\frac{z-5}{2}$ and $\frac{x-4}{1}=\frac{y-1}{2}=\frac{z+3}{0}$ is

Let the position vectors of the points $A,B,C$ and $D$ be $5\hat{i}+5\hat{j}+2\lambda \hat{k},\hat{i}+2\hat{j}+3\hat{k},-2\hat{i}+\lambda \hat{j}+4\hat{k}$ and $-\hat{i}+5\hat{j}+6\hat{k}.$ Let the set $S={\lambda \in \mathbb{R}:$ the points $A,B,C$ and $D$ are coplanar$}.$ The $\underset{\lambda \in S}{\sum }(\lambda +2{)}^{2}$ is equal to

Shortest distance between the lines $\frac{x-1}{2}=\frac{y+8}{-7}=\frac{z-4}{5}\text{ and }\frac{x-1}{2}=\frac{y-2}{1}=\frac{z-6}{-3}\text{ is }$

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: $(A)$ : $\mathrm{max}{|{a}_{1}|,|{a}_{2}|,|{a}_{3}|}\leq |\vec{a}|$ $(B)$ : $|\vec{a}|\leq 3\mathrm{max}{|{a}_{1}|,|{a}_{2}|,|{a}_{3}|}$

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

Let the co-ordinates of one vertex of $\Delta ABC$ be $A(0,2,\alpha )$ and the other two vertices lie on the line $\frac{x+\alpha }{5}=\frac{y-1}{2}=\frac{z+4}{3}$. For $\alpha \in \mathbb{Z}$, if the area of $\Delta ABC$ is $21\mathrm{sq}.\mathrm{units}$ and the line segment $BC$ has length $2\sqrt{21}$ units, then ${\alpha }^{2}$ is equal to _______.

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

The scalar triple product [i j k] is:

Let $a$ and $b$ be two unit vectors such that $|(a+b)+2(a\times b)|=2$. If $\theta \in (0,\pi )$ is the angle between $\hat{a}$ and $\hat{b}$, then among the statements: $(S1):2|\hat{a}\times \hat{b}|=|\hat{a}-\hat{b}|$ $(S2)$ : The projection of $\hat{a}$ on $(\hat{a}+\hat{b})$ is $\frac{1}{2}$

Let $\hat{a}$ and $\hat{b}$ be two unit vectors such that the angle between them is $\frac{\pi }{4}$. If $\theta$ is the angle between the vectors $(\hat{a}+\hat{b})$ and $(\hat{a}+2\hat{b}+2(\hat{a}\times \hat{b}))$ then the value of $164{\mathrm{cos}}^{2}\theta$ is equal to

The line of shortest distance between the lines $\frac{x-2}{0}=\frac{y-1}{1}=\frac{z}{1}$ and $\frac{x-3}{2}=\frac{y-5}{2}=\frac{z-1}{1}$ makes an angle of ${\mathrm{sin}}^{-1}(\sqrt{\frac{2}{27}})$ with the plane $P:ax-y-z=0,(a>0)$. If the image of the point $(1,1,-5)$ in the plane $P$ is $(\alpha ,\beta ,\gamma )$, then $\alpha +\beta -\gamma$ is equal to _____ .

Let $\vec{a}=\hat{i}-\hat{j}+2\hat{k}$ and let $\vec{b}$ be a vector such that $\vec{a}\times \vec{b}=2\hat{i}-\hat{k}$ and $\vec{a}\cdot \vec{b}=3$. Then the projection of $\vec{b}$ on the vector $\vec{a}-\vec{b}$ is:

If the two lines ${l}_{1}:\frac{x-2}{3}=\frac{y+1}{-2},z=2$ and ${l}_{2}:\frac{x-1}{1}=\frac{2y+3}{\alpha }=\frac{z+5}{2}$ are perpendicular, then an angle between the lines ${l}_{2}$ and ${l}_{3}:\frac{1-x}{3}=\frac{2y-1}{-4}=\frac{z}{4}$ is