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. $15\hat{i} + 20\hat{k}$ 

Given $\overrightarrow{\mathrm{a}}+\overrightarrow{\mathrm{b}}+\overrightarrow{\mathrm{c}}+\overrightarrow{\mathrm{d}}=0$, which one of the following is incorrect?

A scalar quantity is one that:

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

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

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

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