Vector Component

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. A vector has a magnitude $12$. When its tail is at origin, it lies between the positive x - axis and the negative y - axis and makes an angle ${30}^{\circ }$ with the x- axis. It’s y - component is

1. $\frac{6}{\sqrt{3}}\hat{j}$
2. $-\frac{6}{\sqrt{3}}\hat{j}$
3. $6\hat{j}$
4. $-6\hat{j}$

Q. An object kept at $\text{O}$ is in equilibrium under the action of four coplanar forces shown in the figure. Find the magnitudes of $F_1$ and $F_2$.

1. $\frac{20}{\sqrt{3}}\text{N},\frac{4}{\sqrt{3}}\text{N}$
2. $20\sqrt{3}\text{N},4\sqrt{3}\text{N}$
3. $\frac{4}{\sqrt{3}}\text{N},\frac{20}{\sqrt{3}}\text{N}$
4. $4\sqrt{3}\text{N},20\sqrt{3}\text{N}$

Q.

A particle is moving with velocity v = $100m{s}^{-1}$ . If one of the rectangular components of a velocity is $50m{s}^{-1}$. Find the other component of velocity and its angle with the given component of velocity.

1. Magnitude $=100\sqrt{3}m{s}^{-1}$

   Angle $={30}^{\circ }$

2. Magnitude $=120\sqrt{3}m{s}^{-1}$

   Angle $={40}^{\circ }$

3. Magnitude $=50\sqrt{3}m{s}^{-1}$

   Angle $={60}^{\circ }$

4. Magnitude $=80\sqrt{3}m{s}^{-1}$

   Angle $={30}^{\circ }$

Q.

A person in a wheelchair is moving up a ramp constant speed. Their total weight is 900N. The ramp makes an angle ${37}^{\circ }$ with the horizontal. Calculate the component of its weight parallel and perpendicular to the ramp.

1. 540 N and 720 N respectively

2. 420 N and 540 N respectively

3. 320 N and 980 N respectively

4. 680 N and 920 N respectively

Q.

Given the magnitude of vector A to be 6 units. Find the component of A along the x-axis?

1. $3\sqrt{3}$

2. $3\sqrt{3}\hat{i}$

3. $-3\sqrt{3}$

4. $-3\sqrt{3}\hat{i}$

Q. The components of force $F$ parallel and perpendicular to the incline are

1. $(\frac{\sqrt{3}}{2}F,\frac{F}{2})$
2. $(\frac{\sqrt{3}}{2}F,0)$
3. $\frac{F}{2},0$
4. $(\frac{F}{2},\frac{\sqrt{3}}{2}F)$

Q. If $\overrightarrow{A}=4\hat{i}-2\hat{j}+6\hat{k}$ and $\overrightarrow{B}=\hat{i}-2\hat{j}-3\hat{k}$ the angle which $\overrightarrow{A}-\overrightarrow{B}$ makes with x-axis is

1. ${\cos }^{-1}\left(\frac{1}{10}\right)$
2. ${\cos }^{-1}\left(\frac{2}{\sqrt{10}}\right)$
3. ${\cos }^{-1}\left(\frac{1}{\sqrt{10}}\right)$
4. ${\cos }^{-1}\left(\frac{1}{5}\right)$

Q. Magnitude of resultant of two vectors $\overrightarrow{P}$ and $\overrightarrow{Q}$ is equal to the magnitude of $\overrightarrow{P}$. Find the angle between $\overrightarrow{Q}$ and the resultant of $2\overrightarrow{P}$ and $\overrightarrow{Q}$.

1. ${90}^{\circ }$.
2. ${60}^{\circ }$.
3. ${75}^{\circ }$.
4. ${90}^{\circ }$.

Q.

Find component of vector $\overrightarrow{A}$ along the direction of $\overrightarrow{B}$ i.e. A||B and in perpendicular direction of $\overrightarrow{B}$ that is $A_{{\perp }_B}$.

1. A||B = $3\sqrt{3};A\perp B=3$

2. A||B = $\frac{3}{3};A\perp B=3\frac{\sqrt{3}}{2}$

3. A||B = $\frac{3}{2};A\perp B=\frac{1}{2}$

4. A||B = $\frac{3}{\sqrt{3}};A\perp B=3\frac{\sqrt{3}}{2}$

Q. The magnitude of components of a vector $\overrightarrow{a}$ along x and y axis respectively are

1. $\text{a cos}\theta ,\text{a sin}\theta$
2. $\text{a sin}\theta ,\text{a cos}\theta$
3. $\text{a sec}\theta ,\text{a sin}\theta$
4. $\text{a tan}\theta ,\text{a cos}\theta$

Q.

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

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

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

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

4. None of these

Q. The magnitude of component of a vector is

1. always less than its magnitude
2. always greater than its magnitude
3. always equal to its magnitude
4. none of these

Q.

A heavy piece of machinery is raised by sliding it a distance of d=10m along a plank oriented at an angle, $\theta ={60}^{\circ }$ as shown in figure. How far has it moved vertically and horizontally?

1. x- 8.66m, y- 5m

2. x - 5m, y- 9m

3. x-5m, y-8.66m

4. x-4.5m, y-8.66m

Q. Magnitude of resultant of two vectors $\overrightarrow{P}$ and $\overrightarrow{Q}$ is equal to the magnitude of $\overrightarrow{P}$. Find the angle between $\overrightarrow{Q}$ and the resultant of $2\overrightarrow{P}$ and $\overrightarrow{Q}$.

1. ${90}^{\circ }$.
2. ${60}^{\circ }$.
3. ${75}^{\circ }$.
4. ${90}^{\circ }$.

Q.

Any vector in an arbitrary direction can always be replaced by two (or three)

1. Parallel vectors which have the original vector as their resultant

2. Mutually perpendicular vectors which have the original vector as their resultant

3. Arbitrary vectors which have the original vector as their resultant

4. It is not possible to resolve a vector

Q.

What angle does a vector make with the positive x-axis if it has a component of 80 units on the negative x-axis and 60 units on the positive y-axis?

1. ${30}^{\circ }$

2. ${53}^{\circ }$

3. ${37}^{\circ }$

4. ${143}^{\circ }$

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 magnitude of component of a vector is

1. always less than its magnitude
2. always greater than its magnitude
3. always equal to its magnitude
4. none of these

Q. Given that $\overrightarrow{C}=\overrightarrow{A}+\overrightarrow{B}$. Also, the magnitude of $\overrightarrow{A},\overrightarrow{B}$ and $\overrightarrow{C}$ are $12,5$ and $13$ units respectively. The angle between $\overrightarrow{A}$ and $\overrightarrow{B}$ is

1. $0^{\circ }$
2. $\frac{\pi }{4}$
3. $\frac{\pi }{2}$
4. $\pi$

Q.

A car is moving in direction $\overrightarrow{r}=-4\hat{i}+3\hat{j}$ with a speed of 10m/s. Write velocity vector of car in vector notation.

1. $-4\hat{i}+3\hat{j}m/s$

2. $-5\hat{i}+8\hat{j}m/s$

3. $-8\hat{i}+6\hat{j}m/s$

4. $-6\hat{i}+8\hat{j}m/s$

Q. If $\overrightarrow{A}=2\hat{i}+3\hat{j}-\hat{k}$ and $\overrightarrow{B}=-\hat{i}+3\hat{j}+4\hat{k}$, then projection of $\overrightarrow{A}$ on $\overrightarrow{B}$ will be

1. $\frac{3}{\sqrt{13}}$
2. $\frac{3}{\sqrt{26}}$
3. $\sqrt{\frac{3}{26}}$
4. $\sqrt{\frac{3}{13}}$

Q. Three forces $\overrightarrow{F_1}$, $\overrightarrow{F_2}$ and $\overrightarrow{F_3}$ act on a body. $\overrightarrow{F_1}$ acts vertically downwards with magnitude $\sqrt{3}\text{N}$ , $\overrightarrow{F_2}$ acts horizontally leftwards with magnitude $1\text{N}$ and $\overrightarrow{F_3}$ acts at an angle $\theta$ with horizontal as shown in the figure. Find the magnitude of $\overrightarrow{F_3}$ and $\theta$ for the body to remain in equilibrium.

1. $2\text{N}$, ${30}^{\circ }$
2. $2\text{N}$, ${60}^{\circ }$
3. $1\text{N}$, ${30}^{\circ }$
4. $1\text{N}$, ${60}^{\circ }$

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. 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. Magnitude of resultant of two vectors $\overrightarrow{P}$ and $\overrightarrow{Q}$ is equal to the magnitude of $\overrightarrow{P}$. Find the angle between $\overrightarrow{Q}$ and the resultant of $2\overrightarrow{P}$ and $\overrightarrow{Q}$.

1. ${90}^{\circ }$.
2. ${60}^{\circ }$.
3. ${75}^{\circ }$.
4. ${90}^{\circ }$.

Q. If $\overrightarrow{A}=3\hat{i}+4\hat{j}and\overrightarrow{B}=6\hat{i}+6\sqrt{3}\hat{j}$, find the angle between $\overrightarrow{A}and\overrightarrow{B}(cos{37}^o=4/5)$

1. ${23}^o$
2. ${16}^o$
3. $7^o$
4. none

Q. The angle which a vector $\hat{i}-\hat{j}+\sqrt{2}\hat{k}$ makes with y - axis is

1. ${60}^{\circ }$
2. ${120}^{\circ }$
3. ${150}^{\circ }$
4. $ta{n}^{-1}(-\frac{1}{2})$

Q. If $\theta$ is angle that vector $\overrightarrow{A}=2\hat{i}+3\hat{j}+5\hat{k}$ makes with x-z plane then

1. $cos\theta =\frac{2}{\sqrt{38}}$
2. $sin\theta =\frac{5}{\sqrt{38}}$
3. $tan\theta =\frac{3}{\sqrt{29}}$
4. $cos\theta =\frac{3}{\sqrt{38}}$

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