B M Sharma Solutions for Chapter: Vectors, Exercise 47: Exercises

According to Newton's second law of motion, the resultant force on a particle is in the direction of acceleration of the particle.

Two forces $F_1$ and $F_2$ are acting on a particle as shown in the figure. The acceleration of the particle is along $x$-axis. Find the value of $F_1$ (in newton).

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 $x+y+z$.

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 $n$.

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 $n$.