The position of a particle in a rectangular co-ordinate system is (3, 2, 5). Then its position vector will be:

1. $5\hat{\mathrm{i}}+6\hat{\mathrm{j}}+2\hat{\mathrm{k}}$

2. $3\hat{\mathrm{i}}+2\hat{\mathrm{j}}+5\hat{\mathrm{k}}$

3. $5\hat{\mathrm{i}}+3\hat{\mathrm{j}}+2\hat{\mathrm{k}}$

4. None of these

A scalar quantity is one that:

If \(\vec{a}\) is a vector and \(x\) is a non-zero scalar, then which of the following is correct?
1. \(x\vec{a}\) is a vector in the direction of \(\vec{a}\)
2. \(x\vec{a}\) is a vector collinear to \(\vec{a}\)
3. \(x\vec{a}\) and \(\vec{a}\) have independent directions
4. \(x\vec{a}\) is a vector perpendicular to \(\vec{a}\)

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

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

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

1. 25 N

2. 75 N

3. 87 N

4. 100 N

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 $\left|\overrightarrow{{\mathrm{v}}_1}+\overrightarrow{{\mathrm{v}}_2}\right|=\left|\overrightarrow{{\mathrm{v}}_1}-\overrightarrow{{\mathrm{v}}_2}\right|$ and $\overrightarrow{{\mathrm{v}}_1}$ and $\overrightarrow{{\mathrm{v}}_2}$ are non-zero vectors, then :

1. $\overrightarrow{{\mathrm{v}}_1}$ is parallel to $\overrightarrow{{\mathrm{v}}_2}$

2. $\overrightarrow{{\mathrm{v}}_1}$= $\overrightarrow{{\mathrm{v}}_2}$

3. $\overrightarrow{{\mathrm{v}}_1}$ and $\overrightarrow{{\mathrm{v}}_2}$ are mutually perpendicular 

4. $\left|\overrightarrow{{\mathrm{v}}_1}\right|=\left|\overrightarrow{{\mathrm{v}}_2}\right|$