Vectors & 3D Geometry PYQ — Page 18

If $\vec{x}=3\hat{i}-6\hat{j}-\hat{k},\vec{y}=\hat{i}+4\hat{j}-3\hat{k}$ and $\vec{z}=3\hat{i}-4\hat{j}-12\hat{k}$, then the magnitude of the projection of $\vec{x}\times \vec{y}$ on $\vec{z}$ is

A vector $\vec{n}$ is inclined to $x$-axis at $45^{\circ}$, to $y$-axis at $60^{\circ}$ and at an acute angle to $z$-axis. If $\vec{n}$ is a normal to a plane passing through the point $(\sqrt{2},-1,1)$ then the equation of the plane is :

The vector $(\hat{i} \times \vec{a} \cdot \vec{b}) \hat{i}+(\hat{j} \times \vec{a} \vec{b}) \hat{j}+(\hat{k} \times \vec{a} \cdot \vec{b}) \hat{k}$ is equal to:

If $\vec{a}$ and $\vec{b}$ are non-collinear vectors, then the value of $\alpha$ for which the vectors $\vec{u}=(\alpha-2) \vec{a}+\vec{b}$ and $\vec{v}=(2+3 \alpha) \vec{a}-3 \vec{b}$ are collinear is :

If the vectors $\vec{\mathrm{AB}}=3\hat{i}+4\hat{k}$ and $\vec{\mathrm{AC}}=5\hat{i}-2\hat{j}+4\hat{k}$ are the sides of a triangle $ABC,$ then the length of the median through $A$ is:

If the lines $\frac{x+1}{2}=\frac{y-1}{1}=\frac{z+1}{3}$ and $\frac{x+2}{2}=\frac{y-k}{3}=\frac{z}{4}$ are coplanar, then the value of $k$ is :

Let $\mathrm{ABC}$ be a triangle with vertices at points $\mathrm{A}$ $(2,3,5)$, B $(-1,3,2)$ and $\mathrm{C}(\lambda, 5, \mu)$ in three dimensional space. If the median through $\mathrm{A}$ is equally inclined with the axes, then $(\lambda, \mu)$ is equal to:

If $\hat{a}, \hat{b}$ and $\hat{c}$ are unit vectors satisfying $\hat{a}-\sqrt{3} \hat{b}+\hat{c}=\overrightarrow{0}$, then the angle between the vectors $\hat{a}$ and $\hat{c}$ is :

Let $\vec{a}=2 \hat{i}+\hat{j}-2 \hat{k}, \vec{b}=\hat{i}+\hat{j}$. If $\vec{c}$ is a vector such that $\vec{a} \bullet \vec{c}=|\vec{c}|,|\vec{c}-\vec{a}|=2 \sqrt{2}$ and the angle between $\vec{a} \times \vec{b}$ and $\vec{c}$ is $30^{\circ}$, then $|(\vec{a} \times \vec{b}) \times \vec{c}|$ equals:

Let $\vec{a}=2 \hat{i}-\hat{j}+\hat{k}, \vec{b}=\hat{i}+2 \hat{j}-\hat{k}$ and $\vec{c}=\hat{i}+\hat{j}-2 \hat{k}$ be three vectors. A vector of the type $\vec{b}+\lambda \vec{c}$ for some scalar $\lambda$, whose projection on $\vec{a}$ is of magnitude $\sqrt{\frac{2}{3}}$ is :

If the lines $\frac{x-1}{2}=\frac{y+1}{3}=\frac{z-1}{4}$ and $\frac{x-3}{1}=\frac{y-k}{2}=\frac{z}{1}$ intersect, then $k$ is equal to

If $\vec{a}=\hat{i}-2 \hat{j}+3 \hat{k}, \vec{b}=2 \hat{i}+3 \hat{j}-\hat{k}$ and $\vec{c}=\lambda \hat{i}+\hat{j}+(2 \lambda-1 \hat{k})$ are coplanar vectors, then $\lambda$ is equal to

Let $A B C D$ be a parallelogram such that $\overrightarrow{A B}=\overrightarrow{\mathrm{q}}, \overrightarrow{A D}=\vec{p}$ and $\angle B A D$ be an acute angle. If $\vec{r}$ is the vector that coincides with the altitude directed from the vertex $B$ to the side $A D$, then $\vec{r}$ is given by

Let $\hat{a}$ and $\hat{b}$ be two unit vectors. If the vectors $\vec{c}=\hat{a}+2 \hat{b}$ and $\vec{d}=5 \hat{a}-4 \hat{b}$ are perpendicular to each other, then the angle between $\hat{\mathrm{a}}$ and $\hat{\mathrm{b}}$ is

Statement 1: If the points $(1,2,2),(2,1,2)$ and $(2,2, z)$ and $(1,1,1)$ are coplanar, then $z=2$. Statement 2: If the 4 points $P, Q, R$ and $S$ are coplanar, then the volume of the tetrahedron $P Q R S$ is 0.

If $a+b+c=0,|\vec{a}|=3,|\vec{b}|=5$ and $|\vec{c}|=7$, then the angle between $\vec{a}$ and $\vec{b}$ is

Statement 1: The vectors $\vec{a}, \vec{b}$ and $\vec{c}$ lie in the same plane if and only if $\vec{a} \cdot(\vec{b} \times \vec{c})=0$ Statement 2: The vectors $\vec{u}$ and $\vec{v}$ are perpendicular if and only if $\vec{u} \cdot \vec{v}=0$ where $\vec{u} \times \vec{v}$ is a vector perpendicular to the plane of $\vec{u}$ and $\vec{v}$

$A B C D$ is parallelogram. The position vectors of $A$ and $C$ are respectively, $3 \hat{i}+3 \hat{j}+5 \hat{k}$ and $\hat{i}-5 \hat{j}-5 \hat{k}$. If $M$ is the midpoint of the diagonal $D B$, then the magnitude of the projection of $\overrightarrow{O M}$ on $\overrightarrow{O C}$, where $O$ is the origin, is

If $\vec{u}=\hat{j}+4 \hat{k}, \vec{v}=\hat{i}+3 \hat{k}$ and $\vec{w}=\cos \theta \hat{i}+\sin \theta \hat{j}$ are vectors in 3-dimensional space, then the maximum possible value of $|\vec{u} \times \vec{v} \cdot \vec{w}|$ is

Statement 1: The shortest distance between the lines $\frac{x}{2}=\frac{y}{-1}=\frac{z}{2}$ and $\frac{x-1}{4}=\frac{y-1}{-2}=\frac{z-1}{4}$ is $\sqrt{2}$. Statement 2: The shortest distance between two parallel lines is the perpendicular distance from any point on one of the lines to the other line.

A unit vector which is perpendicular to the vector $2 \hat{i}-\hat{j}+2 \hat{k}$ and is coplanar with the vectors $\hat{i}+\hat{j}-\hat{k}$ and $2 \hat{i}+2 \hat{j}-\hat{k}$ is

The coordinates of the foot perpendicular from the point $(1,0,0)$ to the line $$ \frac{x-1}{2}=\frac{y+1}{-3}=\frac{z+10}{8} \text { are } $$

The distance of the point $-\hat{i}+2 \hat{j}+6 \hat{k}$ from the straight line that passes through the point $2 \hat{i}+3 \hat{j}-4 \hat{k}$ and is parallel to the vector $6 \hat{i}+3 \hat{j}-4 \hat{k}$ is

The vector $\vec{a}$ and $\vec{b}$ are not perpendicular and $\vec{c}$ and $\vec{d}$ are two vectors satisfying: $\vec{b} \times \vec{c}=\vec{b} \times \vec{d}$ and $\vec{a} \cdot \vec{d}=0$. Then the vector $\vec{d}$ is equal to