Vector Notation

Q.

How do you find the angle of resultant vector?

Q. The angle made by vector $3\hat{i}+3\hat{k}$ with x-axis & y-axis respectively are

1. ${45}^{\circ },{90}^{\circ }$
2. ${90}^{\circ },{90}^{\circ }$
3. ${45}^{\circ },{45}^{\circ }$
4. ${30}^{\circ },{60}^{\circ }$

Q.

A cart of mass $\mathrm{M}$ has a block of mass m attached to it, the coefficient of friction between the block and the cart is $\mathrm{\mu }$. What is the maximum acceleration of the cart so that the block m does not fall $\left(\mathrm{a}\right)$

1. $\frac{\mathrm{\mu }}{\mathrm{g}}$

2. $\mathrm{\mu g}$

3. $\frac{\mathrm{g}}{\mathrm{\mu }}$

4. $\frac{\mathrm{M\mu g}}{\mathrm{m}}$

Q.

What is the $\cos \left({53}^{°}\right)$?

Q. An observer $O$ is moving with the velocity $3\text{m/s}$ in the direction as shown in the figure. Find the time till the observer will be able to see his image in the plane mirror placed at an angle of ${60}^{\circ }$ with the horizontal.

1. $2\text{sec}$
2. $1\text{sec}$
3. $4\text{sec}$
4. $6\text{sec}$

Q.

Draw a graph from the following data. Draw tangents at x = 2, 4, 6 and 8. Find the slopes of these tangents. Verify that the curve drawn is y=$2x^2$ and the slope of tangent is tan $\theta =\frac{dy}{dx}=4x$.

$\begin{array}{ccccccccccc}& & & & & & & & & & \\ \text{x}& 1& 2& 3& 4& 5& 6& 7& 8& 9& 10\\ \text{y}& 2& 8& 18& 32& 50& 72& 98& 128& 162& 200\end{array}$

Q. The speed of the boat in still water is 20 $m/s$ and the speed of river flowing due east is 10 $m/s$. If the boat is on the southern end and it has to cross the river along the shortest path, then the angle at which it should move with respect to north is

1. 
   30$°$ west
2. 
   0$°$
3. 
   45$°$ west
4. 
   60$°$ west

Q. Find the component of vector $\overrightarrow{A}=(2\stackrel{\wedge }{i}+\stackrel{\wedge }{3j})$ along the direction $(\stackrel{\wedge }{i}+\stackrel{\wedge }{j})$.

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

Q.

In the figure shown, explain the difference between $\overrightarrow{DE}$ and $\overrightarrow{ED}$.

Q. A particle is moving such that its position coordinates $(x,y)$ are
$(2m,3m)$ at time $t=0$,
$(6m,7m)$ at time $t=2s$ and
$(13m,14m)$ at time $t=5s$.
Average velocity vector $\left({\overrightarrow{V}}_{av}\right)$ from $t=0$ to $t=5s$ is:

1. $\frac{1}{5}(13\hat{i}+14\hat{j})$
2. $\frac{7}{3}(\hat{i}+\hat{j})$
3. $2(\hat{i}-\hat{j})$
4. $\frac{11}{5}(\hat{i}+\hat{j})$

Q. If vectors $\overrightarrow{A}=\cos \omega t\hat{i}+\sin \omega t\hat{j}$ and $\overrightarrow{B}=\cos \frac{\omega t}{2}\hat{i}+\sin \frac{\omega t}{2}\hat{j}$ are functions of time, then the value of t at which they are orthogonal to each other is :

1. $t=0$
2. $t=\frac{\pi }{4\omega }$
3. $t=\frac{\pi }{2\omega }$
4. $t=\frac{\pi }{\omega }$

Q. The negative of vectors can also be thought of as equal vectors, with their positive counterparts.

1. True
2. False

Q. The force F is exerted on a particle of mass $5$ kg. If the particle starts from rest from origin and its position after $3$ s is $(4.5\hat{i}+13.5\hat{j})m$, then find $\overrightarrow{F}$.

1. $(5\hat{i}+13.5\hat{j})N$
2. $(5\hat{i}+15\hat{j})N$
3. $(5\hat{i}+13\hat{j})N$
4. $(15\hat{i}+5\hat{j})N$

Q. Let $F\left(x\right)={\int }_x^{x^2+\frac{\pi }{6}}2co{s}^{2}tdt$ for all $xϵR$ and $f:[0,\frac{1}{2}]\to [0,\infty )$ be a continuous function. For $\alpha ϵ[0,\frac{1}{2}]$, if $F^{\prime }\left(a\right)+2$ is the area of the region bounded by $x=0,y=0,y=f\left(x\right)$ and $x=a$, then $f\left(0\right)$ is

Q. Give the derivation for resultant of two vectors and for the angle between resultant and a vector.

Q. Resultant of three non-coplanar non-zero vectors $\overline{a},\overline{b}$ and $\overline{c}$ :

1. always lies in the plane containing $\overline{a}+\overline{b}$
2. always lies in the plane containing $\overline{a}-\overline{b}$
3. can be zero
4. cannot be zero

Q. Negative vectors are delocalised in nature.

1. False
2. True

Q.

Vector $\overrightarrow{A}=5\hat{i}+4\hat{j}+3\hat{k}$ is rotated around Z-axis such that its projection on the x-y plane is rotated through ${180}^{\circ }$.Choose the correct option?

1. No it won't as translation of a vector doesn't change it

2. Yes, New Vector =

3. No, it won't as the angle is

4. Yes, New vector =

Q. Two points $A$ and $B$ in space have the co-ordinates $(2,-1,3)$ and $(4,2,5)$ respectively. Find vector $AB$.

Q. A force $\overrightarrow{F}=6\overrightarrow{i}-8\overrightarrow{j}+10\overrightarrow{k}$ N produces an acceleration of magnitude $1m{s}^{-2}$ in a body, the mass of the body would be:

1. $200kg$
2. $6\sqrt{2}kg$
3. $10\sqrt{2}kg$
4. $20kg$

Q. The angle that the vector $\overrightarrow{A}=2\hat{i}+3\hat{j}$ makes with x-axis is

1. ${\tan }^{-1}\left(3/2\right)$
2. ${\tan }^{-1}\left(2/3\right)$
3. ${\sin }^{-1}\left(2/3\right)$
4. ${\cos }^{-1}\left(3/2\right)$

Q. A vector $\underset{a}{\overset{}{\to }}$ has components 2p and 1 with respect to a reactangular cartesian system. The system is rotated through a certain angle about the origin in the counterclockwise direction . If with respect to the new system, $\underset{a}{\overset{}{\to }}$ has componets p+1 end 1 then.

Q. A vector is inclined at $\frac{\pi }{4}$ rad with x-axis & $\frac{\pi }{3}$ rad with y-axis then find angle of vector with z-axis.

1. $\frac{5\pi }{3}$ rad
2. $\frac{2\pi }{3}$ rad
3. $\frac{\pi }{2}$ rad
4. $\frac{\pi }{4}$ rad

Q. State whether True or False :
Motion in 3-Dimension can't be represented with the help of vectors.

1. True
2. False

Q. Resolve a weight of $10$ N in two directions which are parallel an perpendicular to a slope inclined at $30$ to the horizontal,

Q. The rectangular components of a vector lying in xy plane are (n + 1) and 1. If coordinate system is turned by ${60}^0$. They are n and 3 respectively the value of 'n'.

1. 3
2. 3.5
3. 4
4. 4.5

Q. Draw a graph from the following data. Draw tangents at $x=2,4,6$ and $8$. Find the slopes of these tangents. Verify that the curve drawn is $y=2x^2$ and the slope of tangent is $\tan \theta =\frac{dy}{dx}=4x$.

x	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
y	$2$	$8$	$18$	$32$	$50$	$72$	$98$	$128$	$162$	$200$

Q. An airplane is heading north east at a speed of $141{ms}^{-1}$. The northward component of its velocity is:

1. $141.4{ms}^{-1}$
2. Zero
3. $50{ms}^{-1}$
4. $100{ms}^{-1}$

Q. Vector is represented by directed line segment where length of line gives ___________ of vector.

1. Direction
2. Magnitude
3. Both (A) and (B)
4. None of the above.

Q. Explain the resolution of vectors in two dimensions.