Vectors & 3D Geometry PYQ — Page 17

Let $\vec{u}$ be a vector coplanar with the vectors $\vec{a}=2\hat{i}+3\hat{j}-\hat{k}$ and $\vec{b}= \hat{j}+\hat{k}$ . If $\vec{u}$ is perpendicular to $\vec{a}$ and $\vec{u}\cdot \vec{b}=24$, then ${|\vec{u}|}^{2}$ is equal to:

If the position vectors of the vertices $A, B$ and $C$ of a $\triangle \mathrm{ABC}$ are respectively $4 \hat{i}+7 \hat{j}+8 \hat{k}, 2 \hat{i}+3 \hat{j}+4 \hat{k}$ and $2 \hat{i}+5 \hat{j}+7 \hat{k}$, then the position vector of the point, where the bisector of $\angle A$ meets $B C$ is

If $\vec{a},\vec{b},\vec{c}$ are unit vectors such that $\vec{a}+2\vec{b}+2\vec{c}=\vec{0}$, then $|\vec{a}\times \vec{c}|$ is equal to :

The sum of the intercepts on the coordinate axes of the plane passing through the point $(–2,–2,2)$ and containing the line joining the points $(1,–1,2)$ and $(1,1,1)$ is

If the vector $\vec{b}=3\hat{j}+4\hat{k}$ is written as the sum of a vector $\vec{{b}_{1}}$, parallel to $\vec{a}= \hat{i}+ \hat{j}$ and a vector ${\vec{b}}_{2},$ perpendicular to $\vec{a},$ then ${\vec{b}}_{1}\times {\vec{b}}_{2}$ is equal to :

The area (in sq. units) of the parallelogram whose diagonals are along the vectors $8\hat{i}-6\hat{j}$ and $3\hat{i}+4\hat{j}-12\hat{k},$ is:

The distance of the point $(1, 3,-7)$ from the plane passing through the point $(1,-1,-1)$ , having normal perpendicular to both the lines $\frac{x-1}{1}=\frac{y+2}{-2}=\frac{z-4}{3}$ and $\frac{x-2}{2}=\frac{y+1}{-1}=\frac{z+7}{-1}$ , is:

Given, $\vec{a}=2\hat{i}+\hat{j}-2\hat{k}$ and $\vec{b}= \hat{i}+\hat{j}.$ Let $\vec{c}$ be a vector such that $|\vec{c}- \vec{a}|=3, |(\vec{a}\times \vec{b})\times \vec{c}|=3$ and the angle between $\vec{c}$ and $\vec{a}\times \vec{b}$ be $30^{\circ}$ . Then $\vec{a}\cdot \vec{c}$ is equal to:

Let $\vec{a}, \vec{b}$ and $\vec{c}$ be three unit vectors such that $\vec{a} \times (\vec{b} \times \vec{c})=\frac{\sqrt{3}}{2}(\vec{b} + \vec{c}).$ If $\vec{b}$ is not parallel to $\vec{c}$ , then the angle between $\vec{a}$ and $\vec{b}$ is

In a triangle $ABC$, right angle at vertex $A$, if the position vectors of $A,B$ and $C$ are respectively $3\hat{i}+ \hat{j}- \hat{k}, -\hat{i}+3\hat{j}+p\hat{k}$ and $5\hat{i}+q\hat{j}-4\hat{k}$ , then the point $(p,q)$ lies on a line:

The number of distinct real values of $\lambda$, for which the lines $\frac{x-1}{1}=\frac{y-2}{2}=\frac{z+3}{{\lambda }^{2}}$ and $\frac{x-3}{1}=\frac{y-2}{{\lambda }^{2}}=\frac{z-1}{2}$ , are coplanar is

$ABC$ is a triangle in a plane with vertices $A(2, 3, 5), B(-1, 3, 2)$ and $C(\lambda , 5, \mu )$ . If the median through $A$ is equally inclined to the coordinate axes, then the value of $({\lambda }^{3}+{\mu }^{3}+5)$ is

Let $ABC$ be a triangle whose circumcentre is at $P$. If the position vectors $A,B,C$ and $P$ are $\vec{a},\vec{b},\vec{c}$ and $\frac{\vec{a}+\vec{b}+\vec{c}}{4}$ respectively, then the position vector of the orthocentre of this triangle, is :

The shortest distance between the lines $\frac{x}{2}=\frac{y}{2}=\frac{z}{1}$ and $\frac{x+2}{-1}=\frac{y-4}{8}=\frac{z-5}{4},$ lies in the interval:

If the shortest distance between the line $\frac{x-1}{\alpha }=\frac{y+1}{-1}=\frac{z}{1},(\alpha \neq -1)$, and $x+y+z+1=0=2x-y+z+3$ is $\frac{1}{\sqrt{3}}$,then value of $\alpha$ is :

The shortest distance between the $z$ - axis and the line $x+y+2z-3=0=2x+3y+4z-4,$ is

Let $\vec{a}$and $\vec{b}$be two unit vectors such that $|\vec{a}+\vec{b}|=\sqrt{3}$. If $\vec{c}=\vec{a}+2\vec{b}+(\vec{a}\times \vec{b})$ , then $2|\vec{c}|$ is equal to:

Let $\vec{a},\vec{b}\text{and}\vec{c}$ be three non - zero vectors such that no two of them are collinear and $(\vec{a}\times \vec{b})\times \vec{c}=\frac{1}{3}|\vec{b}||\vec{c}|\vec{a}$. If $\theta$ is the angle between vectors $\vec{b}\text{ and}\vec{c}$, then a value of $\mathrm{sin}\theta$ is

In a parallelogram $ABCD, |\vec{AB}|=a, |\vec{AD}|=b& |\vec{AC}|=c$. $\vec{DB}\cdot \vec{AB}$ has the value:

If $|\vec{c}|^2=60$ and $\vec{c} \times(\hat{i}+2 \hat{j}+5 \hat{k})=\overrightarrow{0}$, then a value of $\vec{c} \cdot(-7 \hat{i}+2 \hat{j}+3 \hat{k})$ is:

If $\hat{x}, \hat{y}$ and $\hat{z}$ are three unit vectors in threedimensional space, then the minimum value of $|\hat{x}+\hat{y}|^2+|\hat{y}+\hat{z}|^2+|\hat{z}+\hat{x}|^2$

If $|\vec{a}|=2,|\vec{b}|=3$ and $|2\vec{a}-\vec{b}|=5,$ then $|2\vec{a}+\vec{b}|$ equals :

The angle between the lines whose direction cosines satisfy the equations $l⁡+m+n=0$ and $l{⁡}^{2}={m}^{2}+{n}^{2}$ is

Equation of the line of the shortest distance between the lines $\frac{ x⁡ }{ 1 } = \frac{ y⁡ }{ - 1 } = \frac{ z⁡ }{ 1 }$ and $\frac{ x⁡ - 1 }{ 0 } = \frac{ y⁡ + 1 }{ - 2 } = \frac{ z⁡ }{ 1 }$ is