Vectors & 3D Geometry PYQ — Page 6

Let three vectors $\overrightarrow{\mathrm{a}}=\alpha \hat{i}+4 \hat{j}+2 \hat{k}, \overrightarrow{\mathrm{b}}=5 \hat{i}+3 \hat{j}+4 \hat{k}, \overrightarrow{\mathrm{c}}=x \hat{i}+y \hat{j}+z \hat{k}$ form a triangle such that $\vec{c}=\vec{a}-\vec{b}$ and the area of the triangle is $5 \sqrt{6}$. If $\alpha$ is a positive real number, then $|\vec{c}|^2$ is equal to:

Let $\vec{a}=4 \hat{i}-\hat{j}+\hat{k}, \vec{b}=11 \hat{i}-\hat{j}+\hat{k}$ and $\vec{c}$ be a vector such that $(\vec{a}+\vec{b}) \times \vec{c}=\vec{c} \times(-2 \vec{a}+3 \vec{b})$. If $(2 \vec{a}+3 \vec{b}) \cdot \vec{c}=1670$, then $|\vec{c}|^2$ is equal to :

Let $(\alpha ,\beta ,\gamma )$ be the foot of perpendicular from the point $(1,2,3)$ on the line $\frac{x+3}{5}=\frac{y-1}{2}=\frac{z+4}{3}$. then $19(\alpha +\beta +\gamma )$ is equal to :

The position vectors of the vertices $A,B$ and $C$ of a triangle are $2\hat{i}-3\hat{j}+3\hat{k},2\hat{i}+2\hat{j}+3\hat{k}$ and $-\hat{i}+\hat{j}+3\hat{k}$ respectively. Let $l$ denotes the length of the angle bisector $\mathrm{AD}$ of $\angle \mathrm{BAC}$ where $D$ is on the line segment $\mathrm{BC}$, then $2{l}^{2}$ equals :

Let $\vec{a},\vec{b}$ and $\vec{c}$ be three non-zero vectors such that $\vec{b}$ and $\vec{c}$ are non-collinear if $\vec{a}+5\vec{b}$ is collinear with $\vec{c},\vec{b}+6\vec{c}$ is collinear with $\vec{a}$ and $\vec{a}+\alpha \vec{b}+\beta \vec{c}=\vec{0}$, then $\alpha +\beta$ is equal to

Let $(\alpha ,\beta ,\gamma )$ be mirror image of the point $(2,3,5)$ in the line $\frac{x-1}{2}=\frac{y-2}{3}=\frac{z-3}{4}$. Then $2\alpha +3\beta +4\gamma$ is equal to

The shortest distance between the lines $\frac{x-3}{2}=\frac{y+15}{-7}=\frac{z-9}{5}$ and $\frac{x+1}{2}=\frac{y-1}{1}=\frac{z-9}{-3}$ is

Let $\vec{a}=\hat{i}+\hat{j}+\hat{k}, \vec{b}=2 \hat{i}+4 \hat{j}-5 \hat{k}$ and $\vec{c}=x \hat{i}+2 \hat{j}+3 \hat{k}, x \in \mathbb{R}$. If $\vec{d}$ is the unit vector in the direction of $\vec{b}+\vec{c}$ such that $\vec{a} \cdot \vec{d}=1$, then $(\vec{a} \times \vec{b}) \cdot \vec{c}$ is equal to

The distance, of the point $(7,-2,11)$ from the line $\frac{x-6}{1}=\frac{y-4}{0}=\frac{z-8}{3}$ along the line $\frac{x-5}{2}=\frac{y-1}{-3}=\frac{z-5}{6}$, is :

Consider three vectors $\vec{a}, \vec{b}, \vec{c}$. Let $|\vec{a}|=2,|\vec{b}|=3$ and $\vec{a}=\vec{b} \times \vec{c}$. If $\alpha \in\left[0, \frac{\pi}{3}\right]$ is the angle between the vectors $\vec{b}$ and $\vec{c}$, then the minimum value of $27|\vec{c}-\vec{a}|^2$ is equal to:

Consider a $\Delta ABC$ where $A(1,3,2),B(-2,8,0)$ and $C(3,6,7)$. If the angle bisector of $\angle BAC$ meets the line $BC$ at $D$, then the length of the projection of the vector $\vec{AD}$ on the vector $\vec{AC}$ is:

Let the line of the shortest distance between the lines ${L}_{1}:\vec{r}=(\hat{i}+2\hat{j}+3\hat{k})+\lambda (\hat{i}-\hat{j}+\hat{k})$ and ${L}_{2}:\vec{r}=(4\hat{i}+5\hat{j}+6\hat{k})+\mu (\hat{i}+\hat{j}-\hat{k})$ intersect ${L}_{1}$ and ${L}_{2}$ at $P$ and $Q$ respectively. If $(\alpha ,\beta ,\gamma )$ is the midpoint of the line segment $PQ$, then $2(\alpha +\beta +\gamma )$ is equal to___________

Let $\overrightarrow{O A}=2 \vec{a}, \overrightarrow{O B}=6 \vec{a}+5 \vec{b}$ and $\overrightarrow{O C}=3 \vec{b}$, where $O$ is the origin. If the area of the parallelogram with adjacent sides $\overrightarrow{\mathrm{OA}}$ and $\overrightarrow{\mathrm{OC}}$ is 15 sq. units, then the area (in sq. units) of the quadrilateral $\mathrm{OABC}$ is equal to :

Let $\vec{a}=2 \hat{i}-3 \hat{j}+4 \hat{k}, \vec{b}=3 \hat{i}+4 \hat{j}-5 \hat{k}$ and a vector $\vec{c}$ be such that $\vec{a} \times(\vec{b}+\vec{c})+\vec{b} \times \vec{c}=\hat{i}+8 \hat{j}+13 \hat{k}$. If $\vec{a} \cdot \vec{c}=13$, then $(24-\vec{b} \cdot \vec{c})$ is equal to_______

The set of all $\alpha$, for which the vectors $\vec{a}=\alpha t \hat{i}+6 \hat{j}-3 \hat{k}$ and $\vec{b}=t \hat{i}-2 \hat{j}-2 \alpha t \hat{k}$ are inclined at an obtuse angle for all $t \in \mathbb{R}$, is

Let $\vec{a}=\hat{i}+\hat{j}+\hat{k},\vec{b}=-\hat{i}-8\hat{j}+2\hat{k}$ and $\vec{c}=4\hat{i}+{c}_{2}\hat{j}+{c}_{3}\hat{k}$ be three vectors such that $\vec{b}\times \vec{a}=\vec{c}\times \vec{a}$. If the angle between the vector $\vec{c}$ and the vector $3\hat{i}+4\hat{j}+\hat{k}$ is $\theta$, then the greatest integer less than or equal to ${\mathrm{tan}}^{2}\theta$ is:

Let $\vec{a}={a}_{i}\hat{i}+{a}_{2}\hat{j}+{a}_{3}\hat{k}$ and $\vec{b}={b}_{1}\hat{i}+{b}_{2}\hat{j}+{b}_{3}\hat{k}$ be two vectors such that $|\vec{a}|=1;\vec{a}\cdot \vec{b}=2$ and $|\vec{b}|=4$. If $\vec{c}=2(\vec{a}\times \vec{b})-3\vec{b}$, then the angle between $\vec{b}$ and $\vec{c}$ is equal to :

Let $\vec{a}$ and $\vec{b}$ be two vectors such that $|\vec{b}|=1$ and $|\vec{b}\times \vec{a}|=2$ Then $|(\vec{b}\times \vec{a})-\vec{b}{|}^{2}$ is equal to

Let $A(2,3,5)$ and $C(-3,4,-2)$ be opposite vertices of a parallelogram $ABCD$ if the diagonal $\vec{BD}=\hat{i}+2\hat{j}+3\hat{k}$ then the area of the parallelogram is equal to

If $\vec{a}=\hat{i}+2\hat{j}+\hat{k},\vec{b}=3(\hat{i}-\hat{j}+\hat{k})$ and $\vec{c}$ be the vector such that $\vec{a}\times \vec{c}=\vec{b}$ and $\vec{a}\cdot \vec{c}=3$, then $\vec{a}\cdot ((\vec{c}\times \vec{b})-\vec{b}-\vec{c})$ is equal to

Let $\mathrm{P}(\alpha, \beta, \gamma)$ be the image of the point $\mathrm{Q}(1,6,4)$ in the line $\frac{x}{1}=\frac{y-1}{2}=\frac{z-2}{3}$. Then $2 \alpha+\beta+\gamma$ is equal to_______

Let $\mathrm{P}(\alpha, \beta, \gamma)$ be the image of the point $\mathrm{Q}(3,-3,1)$ in the line $\frac{x-0}{1}=\frac{y-3}{1}=\frac{z-1}{-1}$ and $\mathrm{R}$ be the point $(2,5,-1)$. If the area of the triangle $P Q R$ is $\lambda$ and $\lambda^2=14 K$, then $K$ is equal to :

Let $P(x, y, z)$ be a point in the first octant, whose projection in the $x y$-plane is the point $Q$. Let $O P=\gamma$; the angle between $O Q$ and the positive $x$-axis be $\theta$; and the angle between $O P$ and the positive $z$-axis be $\phi$, where $O$ is the origin. Then the distance of $P$ from the $x$-axis is

Let $Q$ and $R$ be the feet of perpendiculars from the point $P(a,a,a)$ on the lines $x=y,z=1$ and $x=-y,z=-1$ respectively. If $\angle QPR$ is a right angle, then $12{a}^{2}$ is equal to ________