Vectors

1 μC charge moving with velocity $\vec v=\left(\hat i-2\hat j+3\hat k\right)$ m/s in the region of magnetic field $\vec B=\left(2\hat i+3\hat j-5\hat k\right)$ T. The magnitude of force acting on it is $\sqrt\alpha\times10^{-6}$ N. The value of $\alpha$ is ________.

A 1 kg block subjected to two simultaneous forces $\left(2\hat i+3\hat j+4\hat k\right)$ N and $\left(3\hat i-\hat j-2\hat k\right)$ N is moved a distance of 25 m along $\left(3\hat i-4\hat j\right)$ direction. The work done in this process is ________ J.

$\vec A$ is a vector quantity such that $|\vec A|=$ non-zero constant. Which of the following expression is true for $\vec A$ ?

Which of the following relations is true for two unit vector $\hat A$ and $\hat B$ making an angle $\theta$ to each other ?

If force $\vec F=3\hat i+4\hat j-2\hat k$ acts on a particle having position vector $2\hat i+\hat j+2\hat k$ then, the torque about the origin will be :

The position vector of 1 kg object is $\vec r=\left(3\hat i-\hat j\right)$ m and its velocity $\vec v=\left(3\hat j+\hat k\right)$ ms$^{-1}$. The magnitude of its angular momentum is $\sqrt x$ Nm where $x$ is ______.

Two vectors $\vec A$ and $\vec B$ have equal magnitudes. If magnitude of $\vec A+\vec B$ is equal to two times the magnitude of $\vec A-\vec B$, then the angle between $\vec A$ and $\vec B$ will be :

For a free body diagram shown in the figure, the four forces are applied in the '$x$' and '$y$' directions. What additional force must be applied and at what angle with positive x-axis so that the net acceleration of body is zero ?

If $\vec A=\left(2\hat i+3\hat j-\hat k\right)$ m and $\vec B=\left(\hat i+2\hat j+2\hat k\right)$ m. The magnitude of component of vector $\vec A$ along vector $\vec B$ will be ______ m.

If the projection of $2\hat i+4\hat j-2\hat k$ on $\hat i+2\hat j+\alpha\hat k$ is zero. Then, the value of $\alpha$ will be ______.

Two vectors $\vec A$ and $\vec B$ are defined as $\vec A = a\,\hat i$ and $\vec B = a(\cos\omega t\,\hat i + \sin\omega t\,\hat j)$, where $a$ is a constant and $\omega = \pi/6\ \mathrm{rad\,s^{-1}}$. If $|\vec A + \vec B| = \sqrt3|\vec A - \vec B|$ at time $t = \tau$ for the first time, the value of $\tau$, in seconds, is ______.

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

A vector $\vec A$ is rotated by a small angle Δθ radians (Δθ<<1) to get a new vector $\vec B$. In that case $|\vec B-\vec A|$ is :

A body of mass 5 kg under the action of constant force $\vec F=F_x\hat i+F_y\hat j$ has velocity at $t=0$ s as $\vec v=\left(6\hat i-2\hat j\right)$ m/s and at $t=10$ s as $\vec v=+6\hat j$ m/s. The force $\vec F$ is :

A ray of light travelling in the direction $\frac12\left(\hat i+\sqrt3\hat j\right)$ is incident on a plane mirror. After reflection, it travels along the direction $\frac12\left(\hat i-\sqrt3\hat j\right)$. The angle of incidence is