JEE Main 2012 — Vectors & 3D Geometry

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

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

$A B C D$ 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

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

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

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.

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.

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 $\vec{v}$

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