The position of a particle in a rectangular co-ordinate system is $(3, 2, 5)$. Then its position vector will be:
1. $5\hat i + 6\hat j + 2\hat k$
2. $3\hat i + 2\hat j + 5\hat k$
3. $5\hat i + 3\hat j + 2\hat k$
4. None of these

A scalar quantity is one that:

If $\overrightarrow{a}$ is a vector and $x$ is a non-zero scalar, then which of the following is correct?

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 $\overrightarrow {A}= 2\hat i + 4\hat j- 5\hat k,$ 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. $\frac{2}{\sqrt{45}}, \frac{4}{\sqrt{45}}~\text{and}~\frac{-5}{\sqrt{45}}$
2. $\frac{1}{\sqrt{45}}, \frac{2}{\sqrt{45}}~\text{and}~\frac{3}{\sqrt{45}}$
3. $\frac{4}{\sqrt{45}}, 0~\text{and}~\frac{4}{\sqrt{45}}$
4. $\frac{3}{\sqrt{45}}, \frac{2}{\sqrt{45}}~\text{and}~\frac{5}{\sqrt{45}}$

A force $F$ applied at a $30^\circ$ angle to the $x$-axis has the following $X$ and $Y$ components:
1. $\frac{F}{\sqrt{2}}, F$
2. $\frac{F}{2}, \frac{\sqrt{3}}{2}F$
3. $\frac{\sqrt{3}}{2}F, \frac{1}{2}F$
4. $F , \frac{F}{\sqrt{2}}$

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

1. $100~\text{N}, ~175~\text{N}$
2. $86.6~\text{N}, ~100~\text{N}$
3. $100~\text{N}, ~86.6~\text{N}$
4. $100~\text{N}, ~0~\text{N}$

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 $60^{\circ}$ 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: