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 F applied at a 30° angle to the 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. $\mathrm{F}, \frac{\mathrm{F}}{\sqrt{2}}$

If $\overrightarrow{\mathrm{P}}=\overrightarrow{\mathrm{Q}}$ , then which of the following is NOT correct?

1. $\hat{\mathrm{P}}=\hat{\mathrm{Q}}$

2. $\left|\overrightarrow{\mathrm{P}}\right|=\left|\overrightarrow{\mathrm{Q}}\right|$

3. $\mathrm{P}\hat{\mathrm{Q}}=\mathrm{Q}\hat{\mathrm{P}}$

4. $\overrightarrow{\mathrm{P}}+\overrightarrow{\mathrm{Q}}=\hat{\mathrm{P}}+\hat{\mathrm{Q}}$

There are two force vectors, one of 5N and the other of 12N. At what angle should the two vectors be added to get the resultant vector of 17N, 7N, and 13N, respectively:

1. $0°, 180° \mathrm{and} 90°$

2. $0°, 90° \mathrm{and} 180°$

3. $0°, 90° \mathrm{and} 90°$

4. $180°, 0° \mathrm{and} 90°$

A particle moves from position null to $(11\hat{\mathrm{i}}+11\hat{\mathrm{j}}+15\hat{\mathrm{k}})$ due to a uniform force of $(4\hat{\mathrm{i}}+\hat{\mathrm{j}}+3\hat{\mathrm{k}})$N. If the displacement is in m, then the work done will be: (Given: \(W=\vec{F}.\vec{S}\))

1. 100 J

2. 200 J

3. 300 J

4. 250 J

If for two vectors $\overrightarrow{\mathrm{A}}$ and $\overrightarrow{\mathrm{B}}$, $\overrightarrow{\mathrm{A}}\times \overrightarrow{\mathrm{B}}=0$, then the vectors:

1. are perpendicular to each other.

2. are parallel to each other.

3. act at an angle of $60°.$

4. act at an angle of $30°.$

The angle between vectors $\left(\overrightarrow{\mathrm{A}}\times \overrightarrow{\mathrm{B}}\right)$ and $\left(\overrightarrow{\mathrm{B}}\times \overrightarrow{\mathrm{A}}\right)$ is

1. Zero

2. $\mathrm{\pi }$

3. $\mathrm{\pi }/4$

4. $\mathrm{\pi }/2$

The dot product of two mutual perpendicular vector is:

1. 0

2. 1

3. $\infty$

4. None of the above

Let $\mathrm{\theta }$ be the angle between vectors $\overrightarrow{\mathrm{A}}$ and $\overrightarrow{\mathrm{B}}$. Which of the following figures correctly represents the angle $\mathrm{\theta }$?
1. 

2. 

3.  

4. 

If the sum of two unit vectors is also a unit vector, then the magnitude of their difference and angle between the two given unit vectors is

1. $\sqrt{3}, 60°$

2. $\sqrt{3}, 120°$

3. $\sqrt{2}, 60°$

4. $\sqrt{2}, 120°$