Exercises

The ratio of maximum and minimum magnitudes of the resultant of two vectors $\overrightarrow{\mathrm{a}}$ and $\overrightarrow{\mathrm{b}}$ is $3 : 1$. Now, $\left|\overrightarrow{\mathrm{a}}\right|$ is equal to

The magnetic force on a moving charge is $\overrightarrow{F}=q\overrightarrow{v}\times \overrightarrow{B}$. Here $q$ is electric charge on particle, $v$ is velocity of the particle and $B$ is magnetic field.

A charged particle of charge $1 C$ moving with a velocity $\left(\hat{\mathrm{i}}+\hat{\mathrm{j}}-\hat{\mathrm{k}}\right) \mathrm{m} {\mathrm{s}}^{-1}$ in a magnetic field $B=\left(2\hat{\mathrm{i}}-\hat{\mathrm{j}}+4\hat{\mathrm{k}}\right) T$.

Choose the correct option(s).

In vector $\overrightarrow{A}$ and $\overrightarrow{B}$ be respectively equal to $3\hat{\mathrm{i}}-4\hat{\mathrm{j}}+5\hat{\mathrm{k}}$ and $2\hat{\mathrm{i}}+3\hat{\mathrm{j}}-4\hat{\mathrm{k}}$. The unit vector parallel to $\overrightarrow{A}+\overrightarrow{B}$ is $\frac{1}{\sqrt{27}}\left(5\hat{\mathrm{i}}-\mathrm{a}\hat{\mathrm{j}}+\hat{\mathrm{k}}\right)$. Find the value of $a$.

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 $x+y+z$.

If $\overrightarrow{a}=2\hat{\mathrm{i}}-3\hat{\mathrm{j}}+\hat{\mathrm{k}}$ and $\overrightarrow{b}=x\hat{\mathrm{i}}+\hat{\mathrm{j}}+\hat{\mathrm{k}}$ are mutually perpendicular, Find the value of $x$.

Find the volume of parallelepiped $\left(\text{in} {\mathrm{m}}^3\right)$ whose edges are represented by $a=\left(2\hat{\mathrm{i}}-3\hat{\mathrm{j}}+4\hat{\mathrm{k}}\right) \mathrm{m}$,  $b=\left(\hat{\mathrm{i}}+2\hat{\mathrm{j}}-\hat{\mathrm{k}}\right) \mathrm{m}$ and $c=\left(3\hat{\mathrm{i}}-\hat{\mathrm{j}}+2\hat{\mathrm{k}}\right) \mathrm{m}$. 

If $\overrightarrow{A}=2\hat{\mathrm{i}}-\hat{\mathrm{j}}+\hat{\mathrm{k}}$, $\overrightarrow{B}=\hat{\mathrm{i}}+2\hat{\mathrm{j}}-3\hat{\mathrm{k}}$ and $\overrightarrow{C}=3\hat{\mathrm{i}}-y\hat{\mathrm{j}}+5\hat{\mathrm{k}}$ are coplanar. Find the value of $y$.

A force of $1000 \mathrm{N}$ in a particular direction must be applied to a block. For same reason. It is not possible to apply the force in that direction but two forces can be applied to ${30}^{°}$ and ${45}^{°}$ on either side of it in the same plane containing the given force. If the ratio of magnitude of forces as $\sqrt{n}$. Find the value of $n$.