Addition and Subtraction in Unit Vector Notation

Q.

If the sum of two unit vectors is a unit vector, then magnitude of difference is

1. √2
2. √3
3. 2√2
4. √5

Q.

Magnitude of vector which comes on addition of two vectors, $6\hat{i}+7\hat{j}$ and $3\hat{i}+4\hat{j}$ is

1. $\sqrt{136}$
   

2. $\sqrt{13.2}$
   

3. $\sqrt{202}$
   

4. $\sqrt{160}$

Q.

The unit vector parallel to the resultant of the vectors
$\overrightarrow{A}=4\hat{i}+3\hat{j}+6\hat{k}$ and
$\overrightarrow{B}=-\hat{i}+3\hat{j}-8\hat{k}$ is

1. $\frac{1}{7}(3\hat{i}+6\hat{j}-2\hat{k})$

2. $\frac{1}{7}(3\hat{i}+6\hat{j}+2\hat{k})$

3. $\frac{1}{49}(3\hat{i}+6\hat{j}-2\hat{k})$

4. $\frac{1}{49}(3\hat{i}-6\hat{j}+2\hat{k})$
   

Q. Vector $\overrightarrow{A}$ makes equal angles with x, y and z axis. Value of its components (in terms of magnitude of $\overrightarrow{A}$) will be

1. $\sqrt{3}A$
2. $\frac{A}{\sqrt{3}}$
3. $\frac{A}{\sqrt{2}}$
4. $\frac{\sqrt{3}}{A}$

Q.

If $\overrightarrow{A}=4\hat{i}-3\hat{j}$ and $\overrightarrow{B}=6\hat{i}+8\hat{j}$ then magnitude and direction of $\overrightarrow{A}+\overrightarrow{B}$ will be

1. $25,ta{n}^{-1}\left(\frac{3}{4}\right)$
   

2. $10,ta{n}^{-1}\left(5\right)$

3. $5\sqrt{5},ta{n}^{-1}\left(\frac{1}{2}\right)$

4. $5,ta{n}^{-1}\left(\frac{3}{4}\right)$

Q. The position vectors of points $A,B,C$ and $D$ are $A=3\hat{i}+4\hat{j}+5\hat{k}$, $B=4\hat{i}+5\hat{j}+6\hat{k},C=7\hat{i}+9\hat{j}+3\hat{k}$ and $D=4\hat{i}+6\hat{j}$, then the displacement vectors $\overrightarrow{AB}$ and $\overrightarrow{CD}$ are

1. Perpendicular
2. Parallel
3. Antiparallel
4. Inclined at an angle of ${60}^{\circ }$

Q.

$0.4\stackrel{⏜}{i}+0.8\stackrel{⏜}{j}+c\hat{k}$ represents a unit vector when $c$ is

1. $-0.2$

2. $\sqrt{0.2}$

3. $\sqrt{0.8}$

4. $0$

Q.

What vector must be added to the two vectors $\hat{i}-2\hat{j}+2\hat{k}and2\hat{i}+\hat{j}-\hat{k},$ so that the resultant may be a unit vector along x-axis

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

2. $-2\hat{i}+\overrightarrow{j}-\overrightarrow{k}$
   

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

4. $-2\hat{i}-\overrightarrow{j}-\overrightarrow{k}$

Q.

If $y={\sin }^{-1}\left(\frac{1-x^2}{1+x^2}\right)$, then $\frac{dy}{dx}=$

1. $\frac{-1}{\left(1-x^2\right)}$

2. $\frac{1}{\left(1+x^2\right)}$

3. $\frac{-2}{\left(1+x^2\right)}$

4. $\frac{2}{\left(1-x^2\right)}$

Q.

The magnitude of a given vector with end points (4, -4, 0) and (-2, -2, 0) must be

1. $4$

2. $6$

3. $5\sqrt{2}$

4. $2\sqrt{10}$
   

Q.

The position vector of a particle is determined by the expression
$\overrightarrow{r}=3t^2\hat{i}+4t^2\hat{j}+7\hat{k}$. The distance traversed in first 10 sec is

1. 500 m

2. 300 m

3. 100 m

4. 150 m

Q.

If a particle moves from point P (2, 3, 5) to point Q (3, 4, 5). Its displacement vector be

1. $2\hat{i}+4\hat{j}+6\hat{k}$

2. $\hat{i}+\hat{j}+10\hat{k}$

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

4. $\hat{i}+\hat{j}$

Q.

The angle between the two vectors $\overrightarrow{A}=3\hat{i}+4\hat{j}+5\hat{k}$ and $\overrightarrow{B}=3\hat{i}+4\hat{j}-5\hat{k}$ will be

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

2. $0^{\circ }$

3. ${60}^{\circ }$

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

Q.

If ${\sin }^{-1}\left(x-\frac{x^2}{2}+\frac{x^3}{4}-\dots \infty \right)+{\cos }^{-1}\left(x^2-\frac{x^4}{2}+\frac{x^6}{4}-\dots \infty \right)=\frac{\pi }{2}$ for $0<\left|x\right|<\sqrt{2}$, then $x$ equal

1. $\frac{1}{2}$

2. $1$

3. $-\frac{1}{2}$

4. $-1$

Q. Can resultant of 2 vectors be zero ?

1. Yes, when the 2 vectors are same in magnitude and direction
2. No
3. Yes, when the 2 vectors are same in magnitude but opposite in direction
4. Yes, when the 2 vectors are same in magnitude making an angle of $\frac{2\pi }{3}$ with each other

Q. If $\overrightarrow{A}=2\hat{i}+4\hat{j}-5\hat{k}$ the direction of cosines of the vector
$\overrightarrow{A}$ are

1. $\frac{3}{\sqrt{45}},\frac{2}{\sqrt{45}}$ and $\frac{5}{\sqrt{45}}$
   
2. $\frac{1}{\sqrt{45}},\frac{2}{\sqrt{45}}$ and $\frac{3}{\sqrt{45}}$
   
3. $\frac{2}{\sqrt{45}},\frac{4}{\sqrt{45}}$ and $\frac{5}{\sqrt{45}}$
4. $\frac{4}{\sqrt{45}},0$ and $\frac{4}{\sqrt{45}}$
   

Q.

An object of m kg with speed of v m/s strikes a wall at an angle q and rebounds at the same speed and same angle. The magnitude of the change in momentum of the object will be

1. $0$

2. $2mvcos\theta$

3. $2mvsin\theta$

4. $2mv$

Q. Th position vector of four points $A,B,C$ and $D$ are $\hat{i}+\hat{j}+\hat{k},2\hat{i}+5\hat{j},3\hat{i}+2\hat{j}-3\hat{k},\hat{i}-6\hat{j}-\hat{k}$.
Then select the correct relation.

1. $AB=CD$
2. $AB=\frac{1}{2}CD$
3. $AB=\frac{2}{3}CD$
4. $AB=\frac{1}{4}CD$

Q. A body is at rest under the action of three forces, two of which are $\overrightarrow{F_1}=4\hat{i},\overrightarrow{F_2}=6\hat{j}$, the third force is

1. $4\hat{i}+6\hat{j}$
   
2. $-4\hat{i}+6\hat{j}$
   
3. $4\hat{i}-\hat{j}$
   
4. $-4\hat{i}-6\hat{j}$

Q. The vector that must be added to the vector $\hat{i}-3\hat{j}+2\hat{k}$ and $3\hat{i}+6\hat{j}-7\hat{k}$ so that the resultant vector is a unit vector along the y­­-axis is

1. $4\hat{i}+2\hat{j}+5\hat{k}$
   
2. $-4\hat{i}-2\hat{j}+5\hat{k}$
3. $3\hat{i}+4\hat{j}+5\hat{k}$
   
4. $Nullvector$

Q.

With respect to a rectangular cartesian coordinate system, three vectors are expressed as $\overrightarrow{a}=4\hat{i}-\hat{j},\overrightarrow{b}=-3\hat{i}+2\hat{j}$ and
$\overrightarrow{c}=-\hat{k}$ where $\hat{i},\hat{j},\hat{k}$ are unit vectors, along the X, Y and Z-axis respectively. The unit vectors $\hat{r}$along the direction of sum of these vector is

1. $\hat{r}=\frac{1}{\sqrt{3}}(\hat{i}+\hat{j}-\hat{k})$

2. $\hat{r}=\frac{1}{\sqrt{2}}(\hat{i}+\hat{j}-\hat{k})$

3. $\hat{r}=\frac{1}{\sqrt{3}}(\hat{i}-\hat{j}+\hat{k})$

4. $\hat{r}=\frac{1}{\sqrt{2}}(\hat{i}+\hat{j}+\hat{k})$
   

Q.

There are two force vectors, one of 5 N and other of 12 N at what angle the two vectors be added to get resultant vector of 17 N, 7 N and 13 N respectively

1. $0^{\circ },{90}^{\circ }$ and ${180}^{\circ }$

2. $0^{\circ },{90}^{\circ }$ and ${90}^{\circ }$

3. $0^{\circ },{180}^{\circ }$ and ${90}^{\circ }$

4. ${180}^{\circ },0^{\circ }$ and ${90}^{\circ }$

Q. A satellite of mass m is revolving round the earth at a height R above the surface of earth. If g is gravitational acceleration at surface of earth and R is radius of earth then magnitude of momentum of satellite is

1. $\sqrt{\frac{m^{2}Rg}{2}}$
2. $m\sqrt{Rg}$
3. $\sqrt{\frac{m^{2}Rg}{4}}$
4. $\sqrt{2m^{2}Rg}$

Q. A unit vector parallel to the resultant of the vectors $\overrightarrow{A}=\hat{i}+4\hat{j}-2\hat{k}$ and $\overrightarrow{B}=3\hat{i}-5\hat{j}+\hat{k}$ is

1. $4\hat{i}-\hat{j}-\hat{k}$
2. $-4\hat{i}+\hat{j}+\hat{k}$
3. $\frac{4\hat{i}}{\sqrt{18}}-\frac{\hat{j}}{\sqrt{18}}-\frac{\hat{k}}{\sqrt{18}}$
4. $-\frac{4\hat{i}}{\sqrt{18}}+\frac{\hat{j}}{\sqrt{18}}+\frac{\hat{k}}{\sqrt{18}}$

Q. Explain the dot product of two vector quantities.

Q.

The magnitude of a vector on the addition of two vectors $6\hat{i}+7\hat{j}$ and $3\hat{i}+4\hat{j}$

1. $\sqrt{132}$

2. $\sqrt{136}$

3. $\sqrt{160}$

4. $\sqrt{202}$

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

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

Q. The three vectors
$\overrightarrow{A}=3\hat{i}-2\hat{j}+\hat{k},\overrightarrow{B}=\hat{i}-3\hat{j}+5\hat{k}$ and $\overrightarrow{C}=2\hat{i}+\hat{j}-4\hat{k}$ form

1. an equilateral triangle
2. isosceles triangle
3. a right angled triangle
4. no triangle

Q.

Evaluate: $\tan \left[{\cos }^{-1}\left(\frac{4}{5}\right)+{\tan }^{-1}\left(\frac{2}{3}\right)\right]$

1. $\left(\frac{6}{17}\right)$

2. $\left(\frac{17}{6}\right)$

3. $\left(\frac{7}{16}\right)$

4. $\left(\frac{16}{7}\right)$

Q. The angle between the two vectors $\overrightarrow{A}=3\hat{i}+4\hat{j}+5\hat{k}$ and $\overrightarrow{B}=3\hat{i}+4\hat{j}+5\hat{k}$ is

1. Zero
2. None of these
3. ${90}^{\circ }$
4. ${60}^{\circ }$