Chapter 3

Given, vector, \(\mathbf{A}=\hat{\mathbf{i}}-\hat{\mathbf{j}}+2 \hat{\mathbf{k}}\) and vector \(\mathbf{B}=3 \hat{\mathbf{i}}-3 \hat{\mathbf{j}}+6 \hat{\mathbf{k}}\), then which one of the following statements is true? (a) \(A\) is perpendicular to \(B\) (b) \(A\) is parallel to \(B\) (c) Magnitude of \(A\) is half of that of \(B\) (d) Magnitude of \(B\) is equal to that of \(A\)

Given, \(\theta\) is the angle between \(\mathbf{A}\) and \(\mathbf{B} .\) Then, \(|\mathbf{A} \times \mathbf{B}|\) is equal to (a) \(\sin \theta\) (b) \(\cos \theta\) (c) \(\tan \theta\) (d) \(\cot \theta\)

Given, \(\mathbf{p}=3 \hat{\mathbf{j}}+4 \hat{\mathbf{k}}\) and \(\mathbf{Q}=2 \hat{\mathbf{i}}+5 \hat{\mathbf{k}}\). The magnitude of the scalar product of these vectors, is (a) 20 (b) 23 (c) 26 (d) \(5 \sqrt{33}\)

Assertion Angle between \(\hat{\mathbf{i}}+\hat{\mathbf{j}}\) and \(\hat{\mathbf{i}}\) is \(45^{\circ}\) Reason \(\hat{\mathbf{i}}+\hat{\mathbf{j}}\) is equally include to both \(\hat{\mathbf{i}}\) and \(\hat{\mathbf{j}}\) and the angle between \(\hat{i}\) and \(\hat{j}\) is \(90^{\circ}\).

If \(\mathbf{P} \cdot \mathbf{Q}=0\), then \(|\mathbf{P} \times \mathbf{Q}|\) is (a) \(|\mathrm{P}||\mathrm{Q}|\) (b) zero (c) 1 (d) \(\sqrt{P Q}\)

Assertion The vector \(\frac{1}{\sqrt{3}} \hat{\mathbf{i}}+\frac{1}{\sqrt{3}} \hat{\mathbf{j}}+\frac{1}{\sqrt{3}} \hat{\mathbf{k}}\) is a unit vector. Reason Unit vector is one which has unit magnitude and a given direction.

Given, \(\mathbf{c}=\mathbf{a} \times \mathbf{b}\). The angle which a makes with \(\mathbf{c}\) is (a) \(0^{\circ}\) (b) \(45^{\circ}\) (c) \(90^{\circ}\) (d) \(180^{\circ}\)

If \(\mathbf{P}=2 \hat{\mathbf{i}}-3 \hat{\mathbf{j}}+\hat{\mathbf{k}}\) and \(\mathbf{Q}=3 \hat{\mathbf{i}}-2 \hat{\mathbf{j}}\), then \(\mathbf{P} \cdot \mathbf{Q}\) is (a) zero (b) 6 (c) 12 (d) 15

Assertion A vector A points vertically upwards and \(\mathbf{B}\) points towards north. The vector product \(\mathbf{A} \times \mathbf{B}\) is along east. Reason The direction of \(\mathbf{A} \times \mathbf{B}\) is given by right hand rule.

What is the unit vector along \(\hat{\mathbf{i}}+\hat{\mathbf{j}}\) ? (a) \(\frac{\hat{\mathrm{i}}+\hat{\mathrm{j}}}{\sqrt{2}}\) (b) \(\sqrt{2}(\hat{i}+\hat{j})\) (c) \(\hat{\mathrm{i}}+\hat{\mathrm{j}}\) (d) \(\hat{\mathrm{k}}\)

Given, \(\mathbf{C}=\mathbf{A} \times \mathbf{B}\) and \(\mathbf{D}=\mathbf{B} \times \mathbf{A}\). What is the angle between \(\mathbf{C}\) and \(\mathbf{D}\) ? [WB JEE 2009] (a) \(30^{\circ}\) (b) \(60^{\circ}\) (c) \(90^{\circ}\) (d) \(180^{\circ}\)

The adjacent sides of a parallelogram are represented by co-initial vectors \(2 \hat{\mathbf{i}}+3 \hat{\mathbf{j}}\) and \(\hat{\mathbf{i}}+4 \hat{\mathbf{j}}\) The area of the parallelogram is (a) 5 units along \(z\)-axis (b) 5 units in \(x-y\) plane (c) 3 units in \(x-z\) plane (d) 3 units in \(y-z\) plane

\(\mathbf{A}\) and \(\mathbf{B}\) are two vectors given by \(\mathbf{A}=2 \hat{\mathbf{i}}+3 \hat{\mathbf{j}}\) and \(\mathbf{B}=\hat{\mathbf{i}}+\hat{\mathbf{j}}\). The magnitude of the components of \(\mathbf{A}\) and \(\mathbf{B}\) is (a) \(\frac{5}{\sqrt{2}}\) (b) \(\frac{3}{\sqrt{2}}\) (c) \(\frac{7}{\sqrt{2}}\) (d) \(\frac{1}{\sqrt{2}}\)

The magnitudes of the two vectors \(\mathbf{a}\) and \(\mathbf{b}\) are \(a\) and \(b\), respectively. The vector product of a and \(\mathbf{b}\) cannot be (a) equal to zero (b) less than \(a b\) (c) equal to \(a b\) (d) greater than \(a b\)

A particle has an initial velocity of \(3 \hat{\mathbf{i}}+4 \hat{\mathbf{j}}\) and acceleration of \(0.4 \hat{\mathbf{i}}+0.3 \hat{\mathbf{j}}\). Its speed after \(10 \mathrm{~s}\) is (a) \(7 \sqrt{2}\) units (b) 7 units (c) \(8.5\) units (d) 10 units

Given, \(\mathbf{r}=4 \hat{\mathbf{j}}\) and \(\mathbf{p}=2 \hat{\mathbf{i}}+3 \hat{\mathbf{j}}+\hat{\mathbf{k}}\). The angular momentum is (a) \(4 \hat{\mathbf{i}}-8 \hat{k}\) (b) \(8 \hat{i}-4 \hat{k}\) (c) \(8 \hat{\mathrm{j}}\) (d) \(9 \hat{\mathrm{k}}\)

A mass of \(10 \mathrm{~kg}\) is suspended from a spring balance. It is pulled by a horizontal string so that it makes an angle of \(60^{\circ}\) with the vertical. The new reading of the balance is \([\) Karnataka CET 2008] (a) \(10 \sqrt{3} \mathrm{~kg}-\mathrm{wt}\) (b) \(20 \sqrt{3} \mathrm{~kg}-\mathrm{wt}\) (c) \(20 \mathrm{~kg}-\mathrm{wt}\) (d) \(10 \mathrm{~kg}-\mathrm{wt}\)

A force of \((10 \hat{\mathbf{i}}-3 \hat{\mathbf{j}}+6 \hat{\mathbf{k}}) \mathbf{N}\) acts on a body of mass \(100 \mathrm{~g}\) and displaces it from \((6 \hat{\mathbf{i}}+5 \hat{\mathbf{j}}-3 \hat{\mathbf{k}}) \mathrm{m}\) to \((10 \hat{\mathbf{i}}-2 \hat{\mathbf{j}}+7 \hat{\mathbf{k}}) \mathrm{m}\). The work done is (a) \(21 \mathrm{~J}\) (b) \(121 \mathrm{~J}\) (c) \(361 \mathrm{~J}\) (d) \(1000 \mathrm{~J}\)

The component of vector \(\mathbf{A}=a_{x} \hat{\mathbf{i}}+a_{y} \hat{\mathbf{j}}+a_{z} \mathbf{\mathbf { k }}\) along the direction of \((\hat{\mathbf{i}}-\hat{\mathbf{j}})\) is \(\quad\) [EAMCET 2008] (a) \(\left(a_{x}-a_{y}+a_{z}\right)\) (b) \(\left(a_{x}+a_{y}\right)\) (c) \(\left(a_{x}-a_{y}\right) / \sqrt{2}\) (d) \(\left(a_{x}-a_{y}+a_{z}\right)\)

A force, \(\mathbf{F}=2 \hat{\mathbf{i}}+2 \hat{\mathbf{j}} \mathbf{N}\) displaces a particle through \(\mathbf{S}=2 \hat{\mathbf{i}}+2 \hat{\mathbf{k}} \mathbf{m}\) in \(16 \mathrm{~s}\). The power developed by \(\mathbf{F}\) is (a) \(0.25 \mathrm{~J} \mathrm{~s}^{-1}\) (b) \(25 \mathrm{~J} \mathrm{~s}^{-1}\) (c) \(225 \mathrm{~J} \mathrm{~s}^{-1}\) (d) \(450 \mathrm{~J} \mathrm{~s}^{-1}\)

The angle subtended by the vector, \(\mathbf{A}=4 \hat{\mathbf{i}}+3 \hat{\mathbf{j}}+12 \hat{\mathbf{k}}\) with the \(x\)-axis is (a) \(\sin ^{-1}\left(\frac{3}{13}\right)\) (b) \(\sin ^{-1}\left(\frac{4}{13}\right)\) (c) \(\cos ^{-1}\left(\frac{4}{13}\right)\) (d) \(\cos ^{-1}\left(\frac{3}{13}\right)\)

A pendulum of length \(1 \mathrm{~m}\) is released from \(\theta=60^{\circ}\). The rate of change of speed of the bob at \(\theta=30^{\circ}\), is \(\left(g=10 \mathrm{~ms}^{-2}\right)\) (a) \(10 \mathrm{~ms}^{-2}\) (b) \(7.5 \mathrm{~ms}^{-2}\) (c) \(5 \mathrm{~ms}^{-2}\) (d) \(5 \sqrt{3} \mathrm{~ms}^{-2}\)

Two vectors \(\mathbf{a}\) and \(\mathbf{b}\) are such that \(|\mathbf{a}+\mathbf{b}|=|\mathbf{a}-\mathbf{b}|\) What is the angle between a and \(\mathbf{b}\) ? (a) \(0^{\circ}\) (b) \(90^{\circ}\) (c) \(60^{\circ}\) (d) \(180^{\circ}\)

A particle is displaced from a position \((2 \hat{\mathbf{i}}-\hat{\mathbf{j}}-\hat{\mathbf{k}})\) to another position \((3 \hat{\mathbf{i}}+2 \hat{\mathbf{j}}-2 \hat{\mathbf{k}})\) under the action of the force \((2 \hat{\mathbf{i}}+\hat{\mathbf{j}}-\hat{\mathbf{k}})\). The work done by the force is an arbitrary unit is (a) 8 (b) 10 (c) 12 (d) 16

Given, \(\mathbf{A}=4 \hat{\mathbf{i}}+6 \hat{\mathbf{j}}\) and \(\mathbf{B}=2 \hat{\mathbf{i}}+3 \hat{\mathbf{j}}\). Which of the following is correct? (a) \(\mathrm{A} \times \mathrm{B}=0\) (b) \(\mathrm{A} \cdot \mathrm{B}=24\) (c) \(\frac{|\mathrm{A}|}{|\mathrm{B}|}=\frac{1}{2}\) (d) \(\mathrm{A}\) and \(\mathrm{B}\) are anti-parallel

If \(\mathbf{A} \cdot \mathbf{B}=0\) and \(\mathbf{A} \times \mathbf{B}=\mathbf{1}\), then \(\mathbf{A}\) and \(\mathbf{B}\) are (a) perpendicular unit vectors (b) parallel unit vectors (c) parallel (d) perpendicular

A plumb line is suspended from a ceiling of a car moving with horizontal acceleration of \(a\). What will be the angle of inclination with vertical? (a) \(\tan ^{-1}(a / g)\) (b) \(\tan ^{-1}(g / a)\) (c) \(\cos ^{-1}(a / g)\) (d) \(\cos ^{-1}(g / a)\)

The torque of a force \(\mathbf{F}=-3 \hat{\mathbf{i}}+\hat{\mathbf{j}}+5 \hat{\mathbf{k}}\) acting at a point is \(\tau .\) If the position vector of the point is \(7 \hat{\mathbf{i}}+3 \hat{\mathbf{j}}+\hat{\mathbf{k}}\), then \(\tau\) is (a) \(7 \hat{\mathrm{i}}-8 \hat{\mathrm{j}}+9 \hat{\mathrm{k}}\) (b) \(14 \hat{\mathrm{i}}-\hat{\mathrm{j}}+3 \hat{\mathrm{k}}\) (c) \(2 \hat{\mathrm{i}}-3 \hat{\mathrm{j}}+8 \hat{\mathrm{k}}\) (d) \(14 \hat{\mathrm{i}}-38 \hat{\mathrm{j}}+16 \hat{\mathrm{k}}\)

A force, \(\mathbf{F}=(5 \hat{\mathbf{i}}+3 \hat{\mathbf{j}}+2 \hat{\mathbf{k}}) \mathbf{N}\) is applied over a particle which displaces it from its origin to the point \(\mathbf{r}=(2 \hat{\mathbf{i}}-\hat{\mathbf{j}}) \mathbf{m}\). The work done on the particle in joule is (a) \(-\underline{7}\) (b) \(+7\) (c) \(+\underline{10}\) (d) \(+13\)

The area of a parallelogram formed by the vectors \(\mathbf{A}=\hat{\mathbf{i}}-2 \hat{\mathbf{j}}+3 \hat{\mathbf{k}}\) and \(\mathbf{B}=3 \hat{\mathbf{i}}-2 \hat{\mathbf{j}}+\mathbf{k}\) as its adjacent sides, is (a) \(8 \sqrt{3}\) units (b) 64 units (c) 32 units (d) \(4 \sqrt{6}\) units

If \(\mathbf{A} \times \mathbf{B}=\mathbf{B} \times \mathbf{A}\), then the angle between \(A\) and \(B\) is (a) \(\pi\) (b) \(\pi / 3\) (c) \(\pi / 2\) (d) \(\pi / 4\)

Given that, \(\mathbf{A}+\mathbf{B}+\mathbf{C}=0 .\) Out of three vectors, two are equal in magnitude and the magnitude of third vector is \(\sqrt{2}\) times that of either of the two having equal magnitude. Then, the angles between vectors are given by (a) \(45^{\circ}, 45^{\circ}, 90^{\circ}\) (b) \(90^{\circ}, 135^{\circ}, 135^{\circ}\) (c) \(30^{\circ}, 60^{\circ}, 90^{\circ}\) (d) \(45^{\circ}, 60^{\circ}, 90^{\circ}\)

The magnitude of the vectors product of two vectors is \(\sqrt{3}\) times their scalar product. The angle between the two vectors is (a) \(90^{\circ}\) (b) \(60^{\circ}\) (c) \(45^{\circ}\) (d) \(30^{\circ}\)

If, \(\mathbf{A}=2 \hat{\mathbf{i}}+3 \hat{\mathbf{j}}+4 \hat{\mathbf{k}}\) and \(\mathbf{B}=4 \hat{\mathbf{i}}+3 \hat{\mathbf{j}}+2 \hat{\mathbf{k}}\), then angle between \(\mathbf{A}\) and \(\mathbf{B}\) is (a) \(\sin ^{-1}\left(\frac{25}{29}\right)\) (b) \(\sin ^{-1}\left(\frac{29}{25}\right)\) (c) \(\cos ^{-1}\left(\frac{25}{29}\right)\) (d) \(\cos ^{-1}\left(\frac{29}{25}\right)\)

Consider the quantities, pressure, power, energy, impulse, gravitational potential, electrical charge, temperature, area. Out of these, the only vector quantities are (a) Impulse, pressure and area (b) Impulse and area (c) Area and gravitational potential (d) Impulse and pressure

Three vectors \(\mathbf{A}, \mathbf{B}\) and \(\mathbf{C}\) satisfy the relation \(\mathbf{A B}=0\) and \(\mathbf{A C}=0 .\) If \(\mathbf{B}\) and \(\mathbf{C}\) are not lying in the same plane, then \(\mathbf{A}\) is parallel to (a) \(\mathrm{B}\) (b) \(\mathrm{C}\) (c) \(\mathrm{B} \times \mathrm{C}\) (d) \(\mathrm{BC}\)

A force of \((7 \hat{i}+6 \hat{k}) \mathrm{N}\) makes a body move on a rough plane with a velocity of \((3 \hat{\mathbf{j}}+4 \hat{\mathrm{k}}) \mathrm{ms}^{-1}\). Calculate the power in watt (a) 24 (b) 34 (c) 21 (d) 45

What is the angle between \((\hat{\mathbf{i}}+2 \hat{\mathbf{j}}+2 \hat{\mathbf{k}})\) and \(\hat{\mathbf{i}}\) ? (a) \(0^{\circ}\) (b) \(\pi / 6\) (c) \(\pi / 3\) (d) None of these

For what value of \(a, \mathbf{A}=2 \hat{\mathbf{i}}+a \hat{\mathbf{j}}+\hat{\mathbf{k}}\) will be perpendicular to \(\mathbf{B}=4 \hat{\mathbf{i}}-2 \hat{\mathbf{j}}-\hat{\mathbf{k}} ?\) (a) 4 (b) zero (c) 3 (d) 1

The sum of two vectors \(\mathbf{A}\) and \(\mathbf{B}\) is at right angles to their difference. Then (a) \(A=B\) (b) \(A=2 B\) (c) \(B=2 A\) (d) A and B have the same direction

A point of application of a force \(\mathbf{F}=5 \hat{\mathbf{i}}-4 \hat{\mathbf{j}}+2 \hat{\mathbf{k}}\) is moved from \(\mathbf{r}_{1}=2 \hat{\mathbf{i}}+7 \hat{\mathbf{j}}+4 \hat{\mathbf{k}}\) to \(\mathbf{r}_{2}=-5 \hat{\mathbf{i}}+2 \hat{\mathbf{j}}+3 \mathbf{k}\) the work done is (a) \(-17\) units (b) \(-22\) units (c) 33 units (d) \(-33\) units

A vector \(\mathbf{F}_{1}\) is along the positive \(Y\)-axis. If its vector product with another vector \(\mathbf{F}_{2}\) is zero, then \(\mathbf{F}_{2}\) could be (a) \(4 \hat{\mathrm{j}}\) (b) \(\hat{\mathrm{j}}+\hat{\mathrm{k}}\) (c) \(\hat{\mathrm{j}}-\hat{\mathrm{k}}\) (d) \(-4 \hat{\mathrm{i}}\)

If the vectors \(\mathbf{A}=2 \hat{\mathbf{i}}+4 \hat{\mathbf{j}}\) and \(\mathbf{B}=5 \hat{\mathbf{i}}-p \hat{\mathbf{j}}\) are parallel to each other, the magnitude of \(\mathbf{B}\) is (a) \(5 \sqrt{5}\) (b) 10 (c) 15 (d) \(2 \sqrt{5}\)

If the magnitudes of scalar and vector products of two vectors are 6 and \(6 \sqrt{3}\) respectively, then the angle between two vectors is (a) \(15^{\circ}\) (b) \(30^{\circ}\) (c) \(60^{\circ}\) (d) \(75^{\circ}\)

What is the angle between \(\hat{\mathbf{i}}+\hat{\mathbf{j}}+\hat{\mathbf{k}}\) and \(\hat{\mathbf{i}}\) ? (a) \(0^{\circ}\) (b) \(\pi / 6\) (c) \(\pi / 3\) (d) None of these

An object moves along a straight line path from \(P\) to Q under the action of a force \((4 \hat{\mathbf{i}}-3 \hat{\mathbf{j}}+3 \hat{\mathbf{k}}) \mathrm{N}\). If the coordinates of \(P\) and \(Q\) in metres are \((3,3,-1)\) and \((2,-1,4)\) respectively, then the work done by the force is (a) \(+23 \mathrm{~J}\) (b) \(-23 \mathrm{~J}\) (c) \(1015 \mathrm{~J}\) (d) \(\sqrt{35}(4 \hat{\mathrm{i}}-3 \hat{\mathrm{j}}+2 \hat{\mathrm{k}}) \mathrm{J}\)

Given that \(A\) and \(B\) are greater than 1 . The magnitude of \((\mathbf{A} \times \mathbf{B})\) cannot be (a) equal to \(A B\) (b) less than \(A B\) (c) more than \(A B\) (d) equal to \(A / B\)

A force, \(\mathbf{F}=-K(y \hat{\mathbf{i}}+x \hat{\mathbf{j}})\) (where, \(K\) is a positive constant) acts on a particle moving in the \(x y\) plane. Starting from the origin, the particle is taken along the positive \(x\)-axis to the point \((a, 0)\) and then parallel to the \(y\)-axis to the point \((a, a)\). The total work done by the force, \(\mathbf{F}\) on the particle is (a) \(-2 \mathrm{Ka}^{2}\) (b) \(2 \mathrm{Ka}^{2}\) (c) \(-K a^{2}\) (d) \(K a^{2}\)

The coordinates of a moving particle at time \(t\) are given by \(x=c t^{2}\) and \(y=b t^{2}\). The instantaneous speed of the particle is (a) \(2 t(b+c)\) (b) \(2 t(b+c)^{1 / 2}\) (c) \(2 t\left(c^{2}-b^{2}\right)\) (d) \(2 t\left(c^{2}+b^{2}\right)^{1 / 2}\)

Following forces start acting on a particle at rest at the origin of the coordinate system simultaneously \(\mathbf{F}_{1}=5 \hat{\mathbf{i}}-5 \hat{\mathbf{j}}+5 \hat{\mathbf{k}}, \mathbf{F}_{2}=2 \hat{\mathbf{i}}+8 \hat{\mathbf{j}}+6 \hat{\mathbf{k}}, \mathbf{F}_{3}=-6 \hat{\mathbf{i}}+4 \hat{\mathbf{j}}-7 \hat{\mathbf{k}}\) \(\mathbf{F}_{4}=-\hat{\mathbf{i}}-3 \hat{\mathbf{j}}-2 \hat{\mathbf{k}}\). The particle will move (a) in \(x-y\) plane (b) in \(y-z\) plane (c) in \(x-z\) plane (d) along \(x\)-axis