Unit Vectors

Q. Given that $0.2\hat{i}+0.6\hat{j}+a\hat{k}$ is a unit vector. What is the value of a?

1. $\sqrt{0.3}$
2. $\sqrt{0.4}$
3. $\sqrt{0.6}$
4. $\sqrt{0.8}$

Q. If a unit vector is represented by $0.5\hat{i}+0.8\hat{j}+c\hat{k}$ , then the value of $\text{‘}{c}^{\prime }$ is

1. $1$
2. $\sqrt{0.11}$
3. $\sqrt{0.01}$
4. $\sqrt{0.39}$

Q. Find the vector parallel to the vector $\hat{i}-2\hat{j}$ and has magnitude $2\sqrt{5}$ units.

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

Q.

Given $\overrightarrow{A}=0.3\hat{i}+0.4\hat{j}+c\hat{k}$. Calculate the value of c if A is a unit vector.

1. 0.75

2. 0.87

3. 0.96

4. 0.45

Q. If $\overrightarrow{A}$ = 3 $\hat{i}$ + 4 $\hat{j}$, then what is unit vector along $\overrightarrow{A}$ ?

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

Q. A V- shaped current carrying wire is placed inside an uniform magnetic field of magnitude B as shown. Net magnetic force on the wire is

V-आकृति के एक धारावाही तार को दर्शाए अनुसार B परिमाण के एकसमान चुम्बकीय क्षेत्र के अन्दर रखा जाता है। तार पर नेट चुम्बकीय बल है

1. Zero
   
   शून्य
2. 2IBL
3. $\frac{IBL}{2}$
4. IBL

Q. A particle is moving with speed $6m/s$ along the direction of $\overrightarrow{A}=2\hat{i}+2\hat{j}-\hat{k}$, find the velocity (in $m/s$).

1. $(4\hat{i}+4\hat{j}-2\hat{k})$
2. $(2\hat{i}+2\hat{j}-\hat{k})$
3. $(8\hat{i}+8\hat{j}-4\hat{k})$
4. $(\hat{i}+\hat{j}-\hat{k})$

Q. If $\overrightarrow{A}$ = 3 $\hat{i}$ + 4 $\hat{j}$, then what is unit vector along $\overrightarrow{A}$ ?

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

Q. Some equipotential surfaces are shown in figure given below. The magnitude of electric field will be

Q.

Match the following vectors with their correct Notation

$\begin{array}{cc}\begin{array}{cc}\left(i\right)& \text{Vector A}\\ \left(ii\right)& \text{Vector F}\\ \left(iii\right)& \text{Vector D}\\ \left(iv\right)& \text{Vector B}\\ \left(v\right)& \text{Vector E}\end{array}& \begin{array}{cc}\left(A\right)& 3\hat{i}\\ \left(B\right)& -2\hat{j}\\ \left(C\right)& -2\hat{i}\\ \left(D\right)& 2\hat{i}\\ \left(E\right)& 2\hat{i}+2\hat{j}\end{array}\end{array}$

1. (i) - A, (ii) - B, (iii) - C, (iv) - D, (v) - E

2. (i) - D, (ii) - B, (iii) - C, (iv) - A, (v) - E

3. (i) - D, (ii) - B, (iii) - C, (iv) - E, (v) - A

4. (i) - E, (ii) - D, (iii) - C, (iv) - B, (v) - A

Q. Statement -1: The angle between the two vectors $(\hat{i}+\hat{j})$ and $\left(\hat{k}\right)$ is $\frac{\pi }{2}$ radian.
Statement -2: Angle between two vectors $\overrightarrow{A}$ and $\overrightarrow{B}$ is given by $\theta ={\cos }^{-1}\left(\frac{\overrightarrow{A}.\overrightarrow{B}}{|\overrightarrow{A}||\overrightarrow{B}|}\right)$

1. Statement - 1 is True, Statement -2 is True;
   Statment -2 is a correct explanation for Statement -1
2. Statement -1 is True, Statement -2 is True;
   Statement - 2 is NOT a correct explanation for Statement -1
3. Statement - 1 is True, Statement - 2 is False
4. Statement - 1 is False, Statement - 2 is True

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

Q. In the angle subtended by vector $\hat{i}+\hat{j}$ with the x - axis is

1. ${30}^{\circ }$
2. ${45}^{\circ }$
3. ${60}^{\circ }$
4. ${75}^{\circ }$

Q. Unit vector perpendicular to both$\overrightarrow{A}$ and $\overrightarrow{B}$ is

1. $\frac{\overrightarrow{A}\times \overrightarrow{B}}{|\overrightarrow{A}||\overrightarrow{B}|}$
2. $\frac{\overrightarrow{A}\cdot \overrightarrow{B}}{|\overrightarrow{A}||\overrightarrow{B}|\cos \theta }$
3. $\frac{\overrightarrow{A}\times \overrightarrow{B}}{|\overrightarrow{A}||\overrightarrow{B}|\sin \theta }$
4. $\frac{\overrightarrow{A}\cdot \overrightarrow{B}}{|\overrightarrow{A}||\overrightarrow{B}|\sin \theta }$

Q.

If a unit vector is represented by $0.5\hat{i}+0.8\hat{j}+c\hat{k}$, then the value of ‘c’ is

1. 1

2. $\sqrt{0.11}$

3. $\sqrt{0.01}$
   

4. $\sqrt{0.39}$
   

Q.

Given vector $\overrightarrow{A}=2\hat{i}+3\hat{j}$, the angle between $\overrightarrow{A}$ and y-axis is

1. $ta{n}^{-1}\frac{3}{2}$

2. $ta{n}^{-1}\frac{2}{3}$

3. $si{n}^{-1}\frac{2}{3}$

4. $co{s}^{-1}\frac{2}{3}$
   

Q.

Represent point 3, -4, 5 in unit vector representation.

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

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

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

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

Q.

The unit vector along $\hat{i}+\hat{j}$ is

1. $\hat{k}$

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

3. $\frac{\hat{i}+\hat{j}}{\sqrt{2}}$

4. $\frac{\hat{i}+\hat{j}}{2}$

Q.

The sum of vectors A, D and E will be

1. vector A

2. vector E

3. vector D

4. none of these

Q. If $\overrightarrow{A}+\overrightarrow{B}$ is a unit vector along y - axis and $\overrightarrow{A}=\hat{i}-\hat{j}+\hat{k}$, then what is $\overrightarrow{B}$?

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

Q.

Add the following vectors:

$\overrightarrow{A}=5\hat{i}+6\hat{j}-10\hat{k}$

$\overrightarrow{B}=-10\hat{i}-6\hat{j}+10\hat{k}$

1. 0

2. $-5\hat{i}+0\hat{j}+0\hat{k}$

3. $10\hat{i}+6\hat{j}+10\hat{k}$

4. Both a and b

Q. If $\overrightarrow{a}=3\hat{i}+4\hat{j}$ and $\overrightarrow{b}=7\hat{i}+24\hat{j}$, then find the vector having the same magnitude as $\overrightarrow{b}$ and same direction as $\overrightarrow{a}$.

1. $20\hat{i}+15\hat{j}$
2. $15\hat{i}+18\hat{j}$
3. $15\hat{i}+20\hat{j}$
4. $18\hat{i}+15\hat{j}$

Q.

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

Q.

the vector representing the body diagonal from the origin is?

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

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

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

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

Q. Unit vector perpendicular to both$\overrightarrow{A}$ and $\overrightarrow{B}$ is

1. $\frac{\overrightarrow{A}\times \overrightarrow{B}}{|\overrightarrow{A}||\overrightarrow{B}|}$
2. $\frac{\overrightarrow{A}\cdot \overrightarrow{B}}{|\overrightarrow{A}||\overrightarrow{B}|\cos \theta }$
3. $\frac{\overrightarrow{A}\times \overrightarrow{B}}{|\overrightarrow{A}||\overrightarrow{B}|\sin \theta }$
4. $\frac{\overrightarrow{A}\cdot \overrightarrow{B}}{|\overrightarrow{A}||\overrightarrow{B}|\sin \theta }$

Q. The cross product of vectors A = $(\hat{i}+\hat{j})$ and B =$(\hat{i}+\hat{k})$ in

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

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

Q. The unit vector along $\overrightarrow{A}\times \overrightarrow{B}$ is

1. $\frac{\overrightarrow{A}\times \overrightarrow{B}}{|\overrightarrow{A}|\cdot |\overrightarrow{B}|}$
2. $\frac{\overrightarrow{A}\times \overrightarrow{B}}{|\overrightarrow{A}|\cdot |\overrightarrow{B}|\sin \theta }$
3. $\frac{\overrightarrow{A}\times \overrightarrow{B}}{|\overrightarrow{A}|\cdot |\overrightarrow{B}|\cos \theta }$
4. $\frac{\overrightarrow{A}\cdot \overrightarrow{B}}{|\overrightarrow{A}|\cdot |\overrightarrow{B}|\sin \theta }$

Q.

A vector is represented by $3\hat{i}+\hat{j}+2\hat{k}$. Its length in XY plane is

1. $2$

2. $\sqrt{14}$

3. $\sqrt{10}$
   

4. $\sqrt{5}$
   

Q. The unit vector along $\overrightarrow{A}=2\hat{i}+3\hat{j}$ is

1. $2\hat{i}+3\hat{j}$
2. $\frac{2\hat{i}+3\hat{j}}{2}$
3. $\frac{2\hat{i}+3\hat{j}}{3}$
4. $\frac{2\hat{i}+3\hat{j}}{\sqrt{13}}$