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. $R_2=B$

Given that \(\overrightarrow {C}= \overrightarrow {A}+\overrightarrow {B}\)$\mathrm{and}$\(\overrightarrow {C}\) makes an angle $\mathrm{}$\(\alpha\) with \(\overrightarrow {A}\) and \(\beta\) with \(\overrightarrow {B}\). Which of the following options is correct?
1. \(\alpha \) cannot be less than \(\beta\)
2. \(\alpha <\beta, ~\text{if}~A<B\)
3. \(\alpha <\beta, ~\text{if}~A>B\)
4. \(\alpha <\beta, ~\text{if}~A=B\)

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. $\frac{7}{2}(\hat{j}-\hat{i})$

Two forces, \(1\) N and \(2\) N, act along with the lines \(x=0\) and \(y=0\). The equation of the line along which the resultant lies is given by:
1. \(y-2x =0\)
2. \(2y-x =0\)
3. \(y+x =0\)
4. \(y-x =0\)

Two forces of magnitude F have a resultant of the same magnitude F. The angle between the two forces is 

(1) 45°

(2) 120°

(3) 150°

(4) 60°

Two forces with equal magnitudes \(F\) act on a body and the magnitude of the resultant force is \(\frac{F}{3}\). The angle between the two forces is: 
1. \(\cos^{- 1} \left(- \frac{17}{18}\right)\)
2. \(\cos^{- 1} \left(- \frac{1}{3}\right)\)
3. \(\cos^{- 1} \left(\frac{2}{3}\right)\)
4. \(\cos^{- 1} \left(\frac{8}{9}\right)\)$\left(\frac{\mathrm{}}{}\right)$

Two forces are such that the sum of their magnitudes is \(18~\text{N}\) and their resultant is perpendicular to the smaller force and the magnitude of the resultant is \(12~\text{N}\). Then the magnitudes of the forces will be:
1. \(12~\text{N}, 6~\text{N}\)
2. \(13~\text{N}, 5~\text{N}\)
3. \(10~\text{N}, 8~\text{N}\)
4. \(16~\text{N}, 2~\text{N}\)

If two forces of 5 N each are acting along X and Y axes, then the magnitude and direction of resultant is 

(1) $5\sqrt{2},\pi /3$

(2) $5\sqrt{2},\pi /4$

(3) $-5\sqrt{2},\pi /3$

(4) $-5\sqrt{2},\pi /4$