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. \(\overrightarrow{A} + \overrightarrow{B}\)
If \(\left|\overrightarrow A\right|\ne \left|\overrightarrow B\right|\) and \(\left|\overrightarrow A \times \overrightarrow B\right|= \left|\overrightarrow A\cdot \overrightarrow B\right|\), then:
1. | \(\overrightarrow A \perp \overrightarrow B\) |
2. | \(\overrightarrow A ~|| ~\overrightarrow B\) |
3. | \(\overrightarrow A\) is antiparallel to \(\overrightarrow B\) |
4. | \(\overrightarrow A\) is inclined to \(\overrightarrow B\) at an angle of \(45^{\circ}\) |
Two forces of the same magnitude are acting on a body in the East and North directions, respectively. If the body remains in equilibrium, then the third force should be applied in the direction of:
1. North-East
2. North-West
3. South-West
4. South-East
Given are two vectors, \(\overrightarrow{A} = \left(\right. 2 \hat{i} - 5 \hat{j} + 2 \hat{k} \left.\right)\) and \(\overrightarrow{B} = \left(4 \hat{i} - 10 \hat{j} + c \hat{k} \right).\) What should be the value of \(c\) so that vector \(\overrightarrow A \) and \(\overrightarrow B\) would becomes parallel to each other?
1. \(1\)
2. \(2\)
3. \(3\)
4. \(4\)
A block of weight \(W\) is supported by two strings inclined at \(60^{\circ}\) and \(30^{\circ}\) to the vertical. The tensions in the strings are \(T_1\) and \(T_2\) as shown. If these tensions are to be determined in terms of \(W\) using the triangle law of forces, which of these triangles should you draw? (block is in equilibrium):
1. | 2. | ||
3. | 4. |
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 \(\overrightarrow {A} = 2\hat{i} + \hat{j} - \hat{k},\) \(\overrightarrow {B} = \hat{i} + 2\hat{j} + 3\hat{k},\) 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 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
If a vector is inclined at angles \(\alpha ,\beta ,~\text{and}~\gamma\), with \(x\), \(y\), and \(z\)-axis respectively, then the value of \(\sin^{2}\alpha+\sin^{2}\beta+ \sin^{2}\gamma\)
is equal to:
1. \(0\)
2. \(1\)
3. \(2\)
4. \(\frac{1}{2}\)
The current in a circuit is defined as . The charge (q) flowing through a circuit, as a function of time (t), is given by . The minimum charge flows through the circuit at:
1. \(t = 4~\text{s}\)
2. \(t = 2~\text{s}\)
3. \(t = 6~\text{s}\)
4. \(t = 3~\text{s}\)