Addition and Subtraction in Unit Vector Notation

Q.

Can we add 2 vectors of unequal magnitudes and get a zero vector?

1. True

2. False

Q.

If $\tan \left(\frac{A}{2}\right)=\frac{3}{2}$, then $\frac{1+\cos \left(A\right)}{1-\cos \left(A\right)}=$

1. $-5$

2. $5$

3. $\frac{9}{4}$

4. $\frac{4}{9}$

Q.

Vector $\overrightarrow{A}$ is 2 cm long and is ${60}^{\circ }$ above the +ve X axis. Vector $\overrightarrow{A}$ is 2 cm long and is ${60}^{\circ }$ below the x - axis in the fourth quadrant. The value of $\overrightarrow{A}$ + $\overrightarrow{B}$ is

1. 2 cm along -x axis

2. 2 cm along +x axis

3. 4 cm along -x axis

4. 4 cm along +x axis

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

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.

If $m_1,m_2,m_3,m_4$ are respectively the magnitudes of the vectors $a_1=2i-j+k,a_2=3i-4j-4k,a_3=-i+j-k,a_4=-i+3j+k$, then the correct order of $m_1,m_2,m_3,m_4$ is;

1. $m_3<m_1<m_4<m_2$

2. $m_3<m_1<m_2<m_4$

3. $m_3<m_4<m_1<m_2$

4. $m_3<m_4<m_2<m_1$

Q. What displacement must be added to the displacement $25\hat{i}-6\hat{j}m$ to give a displacement of 7.0 m pointing in the x- direction

1. $18\hat{i}-6\hat{j}$
   
2. $32\hat{i}-13\hat{j}$
   
3. $-18\hat{i}+6\hat{j}$
   
4. $-25\hat{i}+13\hat{j}$

Q.

The position vector of a particle is $\overrightarrow{r}=\left(acos\omega t\right)\hat{i}+\left(asin\omega t\right)\hat{j}$. The velocity of the particle is

1. Parallel to the position vector

2. Perpendicular to the position vector

3. Directed towards the origin

4. Directed away from the origin

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 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. 150 m

4. 100 m

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 a vector $\hat{i}+x\hat{j}$ is rotated through an angle $\theta$ and its magnitude is doubled, then it becomes $\hat{i}+4x\hat{j}$. The value of $x$ may be

1. $-\frac{2}{3}$
2. $\frac{1}{2}$
3. $\frac{2}{3}$
4. $2$

Q.

Vectors A and B are $6\hat{i}-7\hat{j}+6\hat{k}$ and $3\hat{i}+4\hat{j}+6\hat{k}$ respectively. Their vector difference in unit vector notation will be

1. $9\hat{i}-3\hat{j}+12\hat{k}$

2. $9\hat{i}-11\hat{j}+12\hat{k}$

3. $10\hat{i}-\hat{j}-\hat{k}$

4. $3\hat{i}-11\hat{j}+0\hat{k}$

Q. The 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.

Add vectors A & D. Which of these is the correct result?

1. $4\hat{i}$

2. $0\hat{i}$

3. $-1\hat{i}$

4. None of these

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. With respect to a rectangular Cartesian co-ordinate 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 $\hat{i},\hat{j}$ and $\hat{k}$ are unit vectors along the x, y and z axes respectively. The unit vector along the direction of sum of these vectors 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. $\hat{r}=\frac{1}{3}(\hat{i}+\hat{j}-\hat{k})$
4. $\hat{r}=\frac{1}{\sqrt{3}}(\hat{i}+\hat{j}+\hat{k})$

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

1. 2 cm along + y - axis
2. $2\sqrt{3}$ cm along + y - axis
3. 1 cm along - x- axis
4. 2 cm along - x - axis

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

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

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

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.

$\overrightarrow{A}=2\hat{i}+\hat{j},3\hat{j}-\hat{k}$ and $\overrightarrow{C}=6\hat{i}-2\hat{k}$. Value of $\overrightarrow{A}-2\overrightarrow{B}+3\overrightarrow{C}$ would be

1. $20\hat{i}-5\hat{j}+4\hat{k}$

2. $20\hat{i}-5\hat{j}-4\hat{k}$

3. $4\hat{i}+5\hat{j}+20\hat{k}$
   

4. $5\hat{i}+4\hat{j}+10\hat{k}$
   

Q.

$\overrightarrow{A}$ is a vector which when added to the resultant of vectors $(2\hat{i}-3\hat{j}+4\hat{k})$ and $(\hat{i}+5\hat{j}+2\hat{k})$ yields a unit vector along the y-axis. Then vector $\overrightarrow{A}$ is

1. $-3\hat{i}-\hat{j}-6\hat{k}$

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

3. $-3\hat{i}-\hat{j}+6\hat{k}$

4. $3\hat{i}+\hat{j}+6\hat{k}$

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.

Given that $\overrightarrow{A}+\overrightarrow{B}=\overrightarrow{C}$ and that $\overrightarrow{C}$ is $\perp$ to $\overrightarrow{A}$. Further if $|\overrightarrow{A}|=|\overrightarrow{C}|$, then what is the angle between $\overrightarrow{A}$ and $\overrightarrow{B}$

1. $\frac{\pi }{4}radian$
   

2. $\frac{\pi }{2}radian$
   

3. $\frac{3\pi }{4}radian$
   

4. $\pi radian$

Q. If a vector $\hat{i}+x\hat{j}$ is rotated through an angle $\theta$ and its magnitude is doubled, then it becomes $\hat{i}+4x\hat{j}$. The value of $x$ may be

1. $-\frac{2}{3}$
2. $\frac{1}{2}$
3. $\frac{2}{3}$
4. $2$