A particle moves from position null to \(\left(11\hat i + 11\hat j + 15\hat k \right)\) due to a uniform force of \(\left(4\hat i + \hat j + 3\hat k\right)\)N. If the displacement is in m, then the work done will be: (Given: \(W=\overrightarrow {F}.\overrightarrow {S}\))
1. \(100~\text{J}\)
2. \(200~\text{J}\)
3. \(300~\text{J}\)
4. \(250~\text{J}\)
The dot product of two mutual perpendicular vector is:
1. \(0\)
2. \(1\)
3. \(\infty\)
4. None of the above
If \(\overrightarrow {A} = 2\hat{i} + \hat{j} - \hat{k},\) \(\overrightarrow {B} = \hat{i} + 2\hat{j} + 3\hat{k},\) and \(\overrightarrow {C} = 6 \hat{i} - 2\hat{j} - 6\hat{k},\) then the angle between \(\left(\overrightarrow {A} + \overrightarrow{B}\right)\) and \(\overrightarrow{C}\) will be:
1. \(30^{\circ}\)
2. \(45^{\circ}\)
3. \(60^{\circ}\)
4. \(90^{\circ}\)
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 \(\overrightarrow{A} = \cos \omega t \hat{i} + \sin \omega t \hat{j}\) and \(\overrightarrow{B} =\cos \frac{\omega t}{2} \hat{i} + \sin \frac{\omega t}{2} \hat{j}\) are functions of time, then the value of \(t\) at which they are orthogonal to each other will be:
1. \(t = \frac{\pi}{2\omega}\)
2. \(t = \frac{\pi}{\omega}\)
3. \(t=0\)
4. \(t = \frac{\pi}{4\omega}\)
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 \(\overrightarrow a= \left(3 \hat{i}+\hat{j}\right) \) and \(\overrightarrow B = \left(2\hat i - \hat j -5\hat k\right)\) is:
1. \(\pm \frac{\left(\right. \hat{i} - 3 \hat{j} + \hat{k} \left.\right)}{\sqrt{11}}\)
2. \(\pm \frac{\left(3 \hat{i} + \hat{j}\right)}{\sqrt{11}}\)
3. \(\pm \frac{\left(\right. 2 \hat{i} - \hat{j} - 5 \hat{k} \left.\right)}{\sqrt{30}}\)
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 \(\overrightarrow {A}\) \(\overrightarrow{B}\) are two vectors inclined to each other at an angle \(\theta,\) then the component of \(\overrightarrow {A}\) perpendicular to \(\overrightarrow {B}\) and lying in the plane containing \(\overrightarrow {A}\) and \(\overrightarrow {B}\) will be:
1. \(\frac{\overrightarrow {A} \overrightarrow{.B}}{B^{2}} \overrightarrow{B}\)
2. \(\overrightarrow{A} - \frac{\overrightarrow{A} \overrightarrow{.B}}{B^{2}} \overrightarrow{B}\)
3. \(\overrightarrow{A} -\overrightarrow{B}\)
4. \(\overrightarrow{A} + \overrightarrow{B}\)
\(\overrightarrow{A}\) and \(\overrightarrow B\) are two vectors and \(\theta\) is the angle between them. If \(\left|\overrightarrow A\times \overrightarrow B\right|= \sqrt{3}\left(\overrightarrow A\cdot \overrightarrow B\right),\) then the value of \(\theta\) will be:
1. | \(60^{\circ}\) | 2. | \(45^{\circ}\) |
3. | \(30^{\circ}\) | 4. | \(90^{\circ}\) |