Application of Vectors

Two vectors $\overrightarrow{A}$ and $\overrightarrow{B}$ are defined as $\overrightarrow{A}=a\hat{\mathrm{i}}$ and $\overrightarrow{B}=a\left(\mathrm{cos\omega t}\hat{\mathrm{i}}+\mathrm{sin\omega t}\hat{\mathrm{j}}\right)$ where $a$ is a constant and $\omega =\frac{\pi }{6} \mathrm{rad} {\mathrm{s}}^{-1}.$ If $|\overrightarrow{A}+\overrightarrow{B}|=\sqrt{3}|\overrightarrow{A}-\overrightarrow{B}|$ at time $t=\tau$ for the first time, the value of $\tau ,$ in seconds is ______.

Three vectors $P, Q$ and $\overrightarrow{R}$ are shown in the figure. Let $S$ be any point on the vector $\overrightarrow{R}.$ The distance between the points $P$ and $S$ is $b\left|\overrightarrow{R}\right|.$ The general relation among vectors $\overrightarrow{P},\overrightarrow{Q}$ and $\overrightarrow{S}$ is:

A projectile is given an initial velocity of $\left(\hat{i}+2\hat{j}\right)m/s$ where $\hat{i}$ is along the ground and $\hat{j}$ is along the vertical. If $g=10 m/s^2,$ the equation of its trajectory is

A particle is moving with velocity $\overrightarrow{v}=K(y\mathrm{i}+x\mathrm{j}),$ where $K$ is a constant. The general equation for its path is

A small particle of mass $m$ is projected at an angle $\mathrm{\theta }$ with the $x$-axis with an initial velocity $v_0$ in the $x-y$ plane as shown in the figure. At a time $t<\frac{v_0\mathrm{sin\theta }}{g},$ the angular momentum of the particle is

Four forces are acting on a particle. One force of magnitude $3 \mathrm{N}$ is directed upward, another is directed $37°$ East of North having the magnitude $5 \mathrm{N}$, third is directed in South-West direction is of magnitude $4\sqrt{2} \mathrm{N}$ and fourth force is $\sqrt{5n} \mathrm{N}$. If the particle is in equilibrium. Find the value of $n$.

The two-point charges $q_1$ and $q_2$ are placed at $\left(0, 0, 0\right)$ and $\left(1, 2, 2\right) \mathrm{m}$, respectively. They repel each other with a force of $3 \mathrm{N}$. The force on the $q_2$ due to $q_1$ is $\overrightarrow{F_{21}}=\left(x\hat{\mathrm{i}}+y\hat{\mathrm{j}}+z\hat{\mathrm{k}}\right) \mathrm{N}$. Find the value of $x+y+z$.

A ray of light is incident along vector $\frac{1}{\sqrt{2}}\hat{\mathrm{i}}-\frac{1}{\sqrt{2}}\hat{\mathrm{j}}+\hat{\mathrm{k}}$ on plane mirror placed in $XY$-plane normal on incidence point is along $Z-$axis.

A force of $\sqrt{3} \mathrm{N}$ makes equal angles with $X-$axis and $Z-$axis. The possible value of force is

If unit vector $\overrightarrow{\mathit{a}}=a_1\hat{\mathrm{i}}+a_2\hat{\mathrm{j}}+a_3\hat{\mathrm{k}}$. The possible values of $a_1$, $a_2$ and $a_3$ are 

Mark the correct option(s).

A particle is in equilibrium in the presence of four forces as shown in the figure, (take $\sqrt{3}=1.7$)
  

If the resultant of several vectors is zero. Then,

In the figure which of the ways indicated for combining the $x$ and $y$ components of a vector are proper to determine that
vector?

Two vectors $\overrightarrow{A}$ and $\overrightarrow{B}$ are drawn from a common point and 

Two vectors $\overrightarrow{A}$ and $\overrightarrow{B}$ lie in one plane. Vector $\overrightarrow{C}$ lies in a different plane. Then $\overrightarrow{A}+\overrightarrow{B}+\overrightarrow{C}$

Two point masses $1$ and $2$ move with uniform velocities ${\overrightarrow{v}}_1$ and ${\overrightarrow{v}}_2$, respectively. Their initial position vectors are ${\overrightarrow{r}}_1$ and ${\overrightarrow{r}}_2$, respectively. Which of the following should be satisfied for the collision of the point masses?

The components of a vector along the $x$- and $y$-directions are $\left(n+1\right)$ and $1$, respectively. If the coordinate system is rotated by $\theta =60°$, then the components change to $n$ and $3$. The value of $n$ is,

A unit vector along the incident ray of light is $\hat{\mathrm{i}}$. The unit vector for the corresponding refracted ray of light is $\hat{\mathrm{r}}.\hat{\mathrm{n}}$, a unit vector normal to the boundary of the medium and directed towards the incident medium. If $\mu$ is the refractive index of the medium, then Snell’s law (second law) of refraction is,

The resultant of three vectors $1$, $2$ and $3$ units whose directions are those of the sides of an equilateral triangle is at an angle of,