A force of \(20\) N acts on a particle along a direction, making an angle of \(60^\circ\) with the vertical. The component of the force along the vertical direction will be:

If $\overrightarrow{\mathrm{A}}=2\hat{\mathrm{i}}+4\hat{\mathrm{j}}-5\hat{\mathrm{k}}$, then the direction cosines of the vector are:

(direction cosines (or directional cosines) of a vector are the cosines of the angles between the vector and the three +ve coordinate axes.)

1. $\frac{2}{\sqrt{45}}, \frac{4}{\sqrt{45}} \mathrm{and} \frac{-5}{\sqrt{45}}$

2. $\frac{1}{\sqrt{45}}, \frac{2}{\sqrt{45}} \mathrm{and} \frac{3}{\sqrt{45}}$

3. $\frac{4}{\sqrt{45}}, 0 \mathrm{and} \frac{4}{\sqrt{45}}$

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

$\overrightarrow{\mathrm{A}}$ is a vector with magnitude A, then the unit vector $\hat{\mathrm{A}}$ in the direction of $\overrightarrow{\mathrm{A}}$ is

1. $\mathrm{A}\overrightarrow{\mathrm{A}}$

2. $\overrightarrow{\mathrm{A}}.\overrightarrow{\mathrm{A}}$

3. $\overrightarrow{\mathrm{A}}\times \overrightarrow{\mathrm{A}}$

4. $\overrightarrow{\mathrm{A}}/\mathrm{A}$

A force \(\mathrm{F}\) applied at a \(30^\circ\) angle to the \(\mathrm{x }\)-axis has the following X and Y components:

1. $\frac{\mathrm{F}}{\sqrt{2}}, \mathrm{F}$

2. $\frac{\mathrm{F}}{2}, \frac{\sqrt{3}}{2}\mathrm{F}$

3. $\frac{\sqrt{3}}{2}\mathrm{F}, \frac{1}{2}\mathrm{F}$

4. $\mathrm{F}, \frac{\mathrm{F}}{\sqrt{2}}$

A child pulls a box with a force of $200 \mathrm{N}$ at an angle of $60°$above the horizontal. Then the horizontal and vertical components of the force will be:
              

1. 100 N, 175 N

2.  86.6 N, 100 N  

3.  100 N, 86.6 N

4. 100 N, 0 N

If  $\overrightarrow{A}=3\hat{i}+4\hat{j}$ and $\overrightarrow{B}=7\hat{i}+24\hat{k}$, then the vector having the same magnitude as that of $\overrightarrow{B}$ and parallel to $\overrightarrow{A}$ is:

1. $15\hat{i} + 20\hat{j}$

2. $\frac{7}{5}\hat{i}+\frac{24}{5}\hat{j}$

3. $20\hat{i}+15\hat{j}$

4. $15\hat{i} + 20\hat{k}$ 

A force is 60 ° inclined to the horizontal. If its rectangular component in the horizontal direction is 50 N, then the magnitude of the force in the vertical direction is

Figure shows the orientation of two vectors u and v in the XY plane.

$\text{ If }\mathrm{u}=\mathrm{a}\hat{\mathrm{i}}+\mathrm{b}\hat{\mathrm{j}}\text{ and }\mathrm{v}=\mathrm{p}\hat{\mathrm{i}}+\mathrm{q}\hat{\mathrm{j}}$

Which of the following is correct?

1. a and p are positive while b and q are negative

2. a, p, and b are positive while q is negative

3. a, q, and b are positive while p is negative

4. a, b, p, and q are all positive

If a unit vector $\hat{j}$ is rotated through an angle of 45° anticlockwise, then the new vector will be:

1.  $\sqrt{2}\hat{\mathrm{i}} + \sqrt{2}\hat{\mathrm{j}}$

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

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

4.  $-\frac{1}{\sqrt{2}}\hat{\mathrm{i}} + \frac{1}{\sqrt{2}}\hat{\mathrm{j}}$