Component of $3\hat{i}+4\hat{j}$ perpendicular to $\hat{i}+\hat{j}$ and in the same plane as that of $3\hat{i}+4\hat{j}$ is:

(1) $\frac{1}{2}(\hat{j}-\hat{i})$

(2) $\frac{3}{2}(\hat{j}-\hat{i})$

(3) $\frac{5}{2}(\hat{j}-\hat{i})$

(4) $\frac{7}{2}(\hat{j}-\hat{i})$

Given that $\stackrel{\rightharpoonup }{\mathrm{C} }= \stackrel{\rightharpoonup }{\mathrm{A}} + \stackrel{\rightharpoonup }{\mathrm{B}} \mathrm{and} \stackrel{\rightharpoonup }{\mathrm{C}}$ makes an angle $\mathrm{\alpha } \mathrm{with} \stackrel{\rightharpoonup }{\mathrm{A}} \mathrm{and} \mathrm{\beta } \mathrm{with} \stackrel{\rightharpoonup }{\mathrm{B}}$. Which of the following options is correct?

1.  $\mathrm{\alpha } \mathrm{cannot} \mathrm{be} \mathrm{less} \mathrm{then} \mathrm{\beta }$

2.  $\mathrm{\alpha } <\mathrm{\beta }, \mathrm{if} \mathrm{A}<\mathrm{B}$

3.  $\mathrm{\alpha } <\mathrm{\beta }, \mathrm{if} \mathrm{A}>\mathrm{B}$

4.  $\mathrm{\alpha } <\mathrm{\beta }, \mathrm{if} \mathrm{A}=\mathrm{B}$

If the angle between the vector $\overrightarrow{A} and \overrightarrow{B}$ is  $\theta$, the value of the product $\left(\overrightarrow{B}\times \overrightarrow{A}\right).\overrightarrow{A}$ is equal to:

1. $B{A}^2 \cos \theta$

2. $B{A}^2 \sin \theta$

3. $B{A}^2 \sin \theta \cos \theta$

4. zero

Two forces A and B have a resultant $R_1$. If B is doubled, the new resultant $R_2$ is perpendicular to A. Then

(1) $R_1=A$

(2) $R_1=B$

(3) $R_2=A$

(4) $R_2=B$

If $|\overrightarrow{u_1}+\overrightarrow{u_2}|=|\overrightarrow{u_1}-\overrightarrow{u_2}| and \left|\overrightarrow{u_2}\right|$ is finite then

1. $\overrightarrow{u_1}$ is parallel to $\overrightarrow{u_2}$

2. $\overrightarrow{u_1}$=$\overrightarrow{u_2}$

3. |$\overrightarrow{u_1}$|=|$\overrightarrow{u_2}$|

4. $\overrightarrow{u_1}$ and $\overrightarrow{u_2}$ are mutually perpendicular

A body is moving under the action of two forces $\overrightarrow{{\mathrm{F}}_1}=2\hat{i}-5\hat{j} , \overrightarrow{{\mathrm{F}}_2}=3\hat{i}-4\hat{j}$. Its velocity will become uniform under an additional third force $\overrightarrow{{\mathrm{F}}_3}$, given by :

1. $5\hat{\mathrm{i}}-\hat{\mathrm{j}}$

2. $-5\hat{\mathrm{i}}-\hat{\mathrm{j}}$

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

4. $-5\hat{\mathrm{i}}+9\hat{\mathrm{j}}$

Two forces are such that the sum of their magnitudes is 18 N and their resultant is perpendicular to the smaller force and magnitude of resultant is 12 N. Then the magnitudes of the forces are 

1. 12 N, 6 N

2. 13 N, 5N

3. 10 N, 8 N

4. 16 N, 2 N

If the magnitude of sum of two vectors is equal to the magnitude of difference of the two vectors, the angle between these vectors is

1. 90°

2. 45° 

3. 180° 

4. 0°

Six vectors $\overrightarrow{a} through \overrightarrow{f}$ have the directions indicated in the figure. Which of the following statements may be true?

(a) $\overrightarrow{b}+\overrightarrow{c}=\overrightarrow{-f}$                        (b) $\overrightarrow{d}+\overrightarrow{c}=\overrightarrow{f}$

(c) $\overrightarrow{d}+\overrightarrow{e}=\overrightarrow{f}$                        (d) $\overrightarrow{b}+\overrightarrow{e}=\overrightarrow{f}$

If a vector $2\hat{\mathrm{i}}+3\hat{\mathrm{j}}+8\hat{\mathrm{k}}$ is perpendicular to the vector $4\hat{\mathrm{j}}-4\hat{\mathrm{i}}+\mathrm{\alpha }\hat{\mathrm{k}}$, then the value of $\mathrm{\alpha }$

1.  $-1$

2.  $-\frac{1}{2}$

3.  $\frac{1}{2}$

4.  $1$