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

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

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 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

The component of vector $\overrightarrow{A}=3\hat{i}+\hat{j}+\hat{k}$ along the direction of $\hat{i}-\hat{j}$ is:

1. $\sqrt{2}$

2. 2

3. $\sqrt{3}$

4. 3

If $\overrightarrow{\mathrm{A}} \mathrm{and} \overrightarrow{\mathrm{B}}$ are two vectors inclined to each other at an angle $\theta$, then the component of $\overrightarrow{\mathrm{A}}$ perpendicular to $\overrightarrow{\mathrm{B}}$ and lying in the plane containing $\overrightarrow{\mathrm{A}} \mathrm{and} \overrightarrow{\mathrm{B}}$ will be:

1.  $\frac{\overrightarrow{\mathrm{A}}\overrightarrow{\mathrm{.B}}}{{\mathrm{B}}^2}\overrightarrow{\mathrm{B}}$

2.  $\overrightarrow{\mathrm{A}} - \frac{\overrightarrow{\mathrm{A}}\overrightarrow{\mathrm{.B}}}{{\mathrm{B}}^2}\overrightarrow{\mathrm{B}}$

3.  $\overrightarrow{\mathrm{A}} - \overrightarrow{\mathrm{B}}$

4.  $\overrightarrow{\mathrm{A}} + \overrightarrow{\mathrm{B}}$

$\overrightarrow{\mathrm{A}}$ and $\overrightarrow{\mathrm{B}}$ are two vectors and θ is the angle between them. If $\left|\overrightarrow{\mathrm{A}}\times \overrightarrow{\mathrm{B}}\right|=\sqrt{3}\left(\overrightarrow{A}.\overrightarrow{B}\right)$, then the value of θ will be:

1. 60^o

2. 45^o

3. 30^o

4. 90^o