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

The dot product of two mutual perpendicular vector is:

1. 0

2. 1

3. $\infty$

4. None of the above

The angle between the two vectors $\left(-2\hat{\mathrm{i}} + 3\hat{\mathrm{j}} + \hat{\mathrm{k}}\right)$ and $\left(\hat{\mathrm{i}} + 2\hat{\mathrm{j}} - 4\hat{\mathrm{k}}\right)$ is:

1. 0°

2. 90°

3. 180°

4. 45°

If \(\vec{A} = 2\hat{i} + \hat{j} - \hat{k}\) ,  \(\vec{B} = \hat{i} + 2\hat{j} + 3\hat{k}\) , and \(\vec{C} = 6 \hat{i} - 2\hat{j} - 6\hat{k}\) , then the angle between \((\vec{A} + \vec{B})\) and \(\vec{C}\) will be

1.  30°

2.  45°

3.  60°

4.  90°

The magnitude of the resultant of two vectors of magnitude 3 units and 4 units is 1 unit. What is the value of their dot product?

1.  –12 units

2.  –7 units

3.  –1 unit

4.  0

$\overrightarrow{\mathrm{A}} \mathrm{and} \overrightarrow{\mathrm{B}}$ are two vectors given by $\overrightarrow{\mathrm{A} }= 2\hat{\mathrm{i}} + 3\hat{\mathrm{j}}$ and $\overrightarrow{\mathrm{B}} = \hat{\mathrm{i}} + \hat{\mathrm{j}}$. The component of $\stackrel{\to \; }{\mathrm{A}}$parallel to $\overrightarrow{\mathrm{B}}$ is:

1.  $\frac{1}{2}\left(2\hat{\mathrm{i}} - \hat{\mathrm{j}}\right)$

2.  $\frac{5}{2}\left(\hat{\mathrm{i}} - \hat{\mathrm{j}}\right)$

3.  $\frac{5}{2}\left(\hat{\mathrm{i}} + \hat{\mathrm{j}}\right)$

4.  $\frac{\left(3\hat{\mathrm{i}} - 2\hat{\mathrm{j}}\right)}{2}$

If vector $\overrightarrow{\mathrm{A}} = \cos \omega t\hat{i} + \sin \omega t\hat{j}$ and $\overrightarrow{\mathrm{B}} = \cos \frac{\omega t}{2}\hat{i} + \sin \frac{\omega t}{2}\hat{j}$ are functions of time, then the value of t at which they are orthogonal to each other will be:

1. $\mathrm{t} = \frac{\mathrm{\pi }}{2\mathrm{\omega }}$

2. $\mathrm{t} = \frac{\mathrm{\pi }}{\mathrm{\omega }}$

3. $\mathrm{t} = 0$

4. $\mathrm{t} = \frac{\mathrm{\pi }}{4\mathrm{\omega }}$

The vector sum of two forces is perpendicular to their vector difference. In that case, the forces:

1. are not equal to each other in magnitude.

2. cannot be predicted.

3. are equal to each other.

4. are equal to each other in magnitude.

The angle which the vector $\overrightarrow{A}=2\hat{i}+3\hat{j}$ makes with the y-axis, where $\hat{i}$ and $\hat{j}$ are unit vectors along x- and y-axis, respectively, is

1. cos^-1 (3/5) 

2. cos^-1 (2/3)

3. tan^-1 (2/3)

4. sin^-1 (2/3)

The unit vector perpendicular to vectors $\overrightarrow{\mathrm{a}}=\left(3\hat{\mathrm{i}}+\hat{\mathrm{j}}\right) \mathrm{and} \overrightarrow{\mathrm{b}}=\left(2\hat{\mathrm{i}}-\hat{\mathrm{j}}-5\hat{\mathrm{k}}\right)$ is

1. $\pm \frac{(\hat{\mathrm{i}}-3\hat{\mathrm{j}}+\hat{\mathrm{k}})}{\sqrt{11}}$

2. $\pm \frac{\left(3\hat{\mathrm{i}}+\hat{\mathrm{j}}\right)}{\sqrt{11}}$

3. $\pm \frac{(2\hat{\mathrm{i}}-\hat{\mathrm{j}}-5\hat{\mathrm{k}})}{\sqrt{30}}$

4. None of these