Vector Component

Q.

The resultant of two vectors $A$ and $B$ is perpendicular to vector $A$ and its magnitude is equal to half the magnitude of vector $B$. The angle between $A$ and $B$ is

1. $120°$

2. $150°$

3. $135°$

4. None of these

Q.

The angle made by the vector $A=\hat{i}+\hat{j}$ with x- axis is

1. ${90}^{\circ }$

2. ${22.5}^{\circ }$

3. ${30}^{\circ }$

4. ${45}^{\circ }$

Q.

Two forces, while acting on a particle in opposite; directions, have the resultant of $10N$. If they act at right angles to each other, the resultant is found to be $50N$. Find the two forces.

Q. A person pushes a box kept on a horizontal surface with a force of $100N$. In unit vector notation, the force can be expressed as

1. $100(\hat{i}+\hat{j})$
2. $100(\hat{i}-\hat{j})$
3. $50\sqrt{2}(\hat{i}-\hat{j})$
4. $50\sqrt{2}(\hat{i}+\hat{j})$

Q. What is the resultant of three coplanar forces $300\text{N}$ at $0^o$, $400\text{N}$ at ${30}^{\circ }$ and $400\text{N}$ at ${150}^{\circ }$ with $x-$ axis?

1. $500\text{N}$
2. $700\text{N}$
3. $1100\text{N}$
4. $300\text{N}$

Q. The vector projection of a vector $3\hat{i}+4\hat{k}$ on $y$ - axis is

1. $5$
2. $4$
3. $3$
4. Zero

Q. The sum of the magnitudes of two forces acting at a point is 18 and the magnitude of their resultant is 12. If the resultant is at 90° with the force of smaller magnitude, what are the magnitudes of forces?

1. $5,12$
2. $12,13$
3. $5,13$
4. $5,5$

Q.

The resultant of two forces $3P$ and $2P$ is $R$, if the first force is doubled, the resultant is also doubled. The angle between the forces is

1. $\frac{\mathrm{\pi }}{3}$

2. $\frac{2\mathrm{\pi }}{3}$

3. $\frac{\mathrm{\pi }}{6}$

4. $\frac{5\mathrm{\pi }}{6}$

Q.

In the given figure two tiny conducting balls of identical mass m and identical charge q hang from non-conducting threads of equal length L. Assume that $\theta$ is so small that $tan\theta \approx sin\theta$, then for equilibrium x is equal to

1. ${\left(\frac{q{L}^2}{2\pi {ϵ}_{0}mg}\right)}^{1/3}$
2. ${\left(\frac{q^{2}L}{2\pi {ϵ}_{0}mg}\right)}^{1/3}$
3. ${\left(\frac{q^2{L}^2}{4\pi {ϵ}_{0}mg}\right)}^{1/3}$
4. ${\left(\frac{q^{2}L}{4\pi {ϵ}_{0}mg}\right)}^{1/3}$

Q. $\overrightarrow{A}$ and $\overrightarrow{B}$ are two vectors given by $\overrightarrow{A}=2\hat{i}+3\hat{j}$ and $\overrightarrow{B}=\hat{i}+\hat{j}$. The magnitude of the component of $\overrightarrow{A}$ along $\overrightarrow{B}$ is

1. $\frac{5}{\sqrt{2}}$
2. $\frac{3}{\sqrt{2}}$
3. $\frac{7}{\sqrt{2}}$
4. $\frac{1}{\sqrt{2}}$

Q.

If $A,B,C$are the angles of a triangle then find $\cos A+\cos B+\cos C$.

Q.

Two forces, F₁ and F₂ are acting on a body. One force is double that of the other force and the resultant is equal to the greater force. Then the angle between the two forces is

1. $A={\cos }^{-1}\left(\frac{1}{2}\right)$

2. $A={\cos }^{-1}\left(\frac{-1}{2}\right)$

3. $A={\cos }^{-1}\left(\frac{-1}{4}\right)$

4. $A={\cos }^{-1}\left(\frac{1}{4}\right)$

Q. What is the maximum number of rectangular components into which a vector can be split?

1. 2
2. 3
3. 4
4. Infinite

Q. Vector $\overrightarrow{A}$ is of length $2\text{cm}$ and makes an angle ${60}^{\circ }$ with the $x$ -axis in the first quadrant. Vector $\overrightarrow{B}$ is of length $2\text{cm}$ and ${60}^{\circ }$ below the $x$-axis in the fourth quadrant. The sum $\overrightarrow{A}+\overrightarrow{B}$ is a vector of magnitude in $\text{cm}$.

1. $2$ along $+y$-axis
2. $2$ along $+x$-axis
3. $1$ along $-x$-axis
4. $2$ along $-x$-axis

Q. Two vectors acting in the opposite directions have a resultant of 10 units. If they act at right angles to each other, then the resultant is 50 units. Calculate the magnitude of two vectors.

1. $40,30$
2. $50,40$
3. $40,50$
4. $30,40$

Q. Component of $3\hat{i}+4\hat{j}$ perpendicular to $\hat{i}+\hat{j}$ and in the same plane as that of $3\hat{i}+4\hat{j}$ is

1. $\frac{1}{2}(-\hat{i}+\hat{j})$
2. $\frac{1}{3}(\hat{i}-\hat{j})$
3. $\frac{1}{4}(-\hat{i}+\hat{j})$
4. $\frac{1}{5}(\hat{i}-\hat{j})$

Q. Two bodies of masses $1$ $kg$ and $3$ $kg$ have position vectors $\hat{i}+2j+k\text{and}-3\hat{i}-2\hat{j}+\hat{k}$ respectively. The centre of mass of this system has a position vector

1. $-2\hat{i}+2\hat{k}$
2. $-2\hat{i}-\hat{j}+\hat{k}$
3. $-\hat{i}+\hat{j}+\hat{k}$
4. $-2\hat{i}-\hat{j}+2\hat{k}$

Q. The angle between the vectors $\overrightarrow{A}$ and $\overrightarrow{B}$ is $\theta$. The triple product $\overrightarrow{A}.(\overrightarrow{B}\times \overrightarrow{A})$ is equal to.

1. $A^{2}Bsi{n}^2\theta$
2. $A^{2}Bsin\theta cos\theta$
3. $A^{2}Bco{s}^2\theta$
4. \N

Q.

The resultant force of $5N$ and $10N$ cannot be

1. $12N$

2. $8N$

3. $4N$

4. $5N$

Q. The resultant of $\overrightarrow{P}$ and $\overrightarrow{Q}$ is perpendicular to $\overrightarrow{P}$. What is the angle between $\overrightarrow{P}$ and $\overrightarrow{Q}$

1. ${\cos }^{-1}\left(\frac{P}{Q}\right)$
2. ${\cos }^{-1}(-\frac{P}{Q})$
3. ${\sin }^{-1}\left(\frac{P}{Q}\right)$
4. ${\sin }^{-1}(-\frac{P}{Q})$

Q.

Which of the following sets of concurrent forces may be in equilibrium

1. 

2. 

3. 

4. 

Q. The angle between the vector $2\hat{i}+4\hat{k}$ and the $Y$- axis is

1. ${45}^{\circ }$
2. ${180}^{\circ }$
3. $0^{\circ }$
4. ${90}^{\circ }$

Q.

Position of a particle in a rectangular-co-ordinate system is (3, 2, 5). Then its position vector will be

1. $5\hat{i}+3\hat{j}+2\hat{k}$
   

2. $3\hat{i}+5\hat{j}+2\hat{k}$

3. $3\hat{i}+2\hat{j}+5\hat{k}$

4. $Noneofthese$

Q. What is the maximum number of components into which a vector can be split?

1. 2
2. 3
3. 4
4. Infinite

Q. A vector in xy - plane has a magnitude of $25m$ and magnitude of x - component is $20m$. The angle it makes with the positive x - axis is

1. ${53}^{\circ }$
2. ${26}^{\circ }$
3. ${29}^{\circ }$
4. ${37}^{\circ }$

Q. As shown in figure, the tension in the horizontal cord is $30\text{N}$. The weight $W$ and tension in the string $OA$ in Newton are

1. $30\sqrt{3},30$
2. $30\sqrt{3},60$
3. $60\sqrt{3},30$
4. None of these

Q.

What is i and j in a vector?

Q. During the swinging of a simple pendulum,

1. The work done by tension force is always zero
2. The work done by the gravitational force is zero
3. The mechanical energy of the bob remains constant in the presence of air resistance
4. The mechanical energy of the bob does not remain constant in the absence of air

Q. The horizontal component of a force of $10N$ inclined at ${30}^o$ to vertical is

1. $3N$
2. $5\sqrt{3}N$
3. $5N$
4. $\frac{10}{\sqrt{3}}N$

Q.

What is the simplest type of motion?