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
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. \(0\)
Identify the unit vector in the following:
1. \(\hat i + \hat j\)
2. \(\frac{\hat i}{\sqrt{2}}\)
3. \(\hat k - \frac{\hat i}{\sqrt{2}}\)
4. \(\frac{\hat i +\hat j}{\sqrt{2}}\)
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. |
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. \(4\)
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
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 \(45^{\circ}\) |
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 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. \(\frac{1}{2}\)