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. \(y-x =0\)

Given that \(\vec {C}= \vec{A}+\vec {B}~\text{and}~\vec{C}\) makes an angle $\mathrm{}$\(\alpha\) with \(\vec{A}\) and\(\beta\)with \(\vec {B}\). 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. \(\alpha <\beta, ~\text{if}~A=B\)

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. $R_2=B$

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. \(\cos^{- 1} \left(\frac{8}{9}\right)\)

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. $-5\sqrt{2},\pi /4$

If the magnitude of the sum of two vectors is equal to the magnitude of the difference between the two vectors, the angle between these vectors is:
1. \(90^{\circ}\)
2. \(45^{\circ}\)
3. \(180^{\circ}\)
4. \(0^{\circ}\)

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

      
1. \(\overrightarrow b + \overrightarrow c = -\overrightarrow f\)
2. \(\overrightarrow d + \overrightarrow c = \overrightarrow f\)
3. \(\overrightarrow d + \overrightarrow e = \overrightarrow f\)
4. \(\overrightarrow b + \overrightarrow e = \overrightarrow f\)