Chapter 3

Two forces, each equal to \(\frac{P}{2}\) act at right angles. Their effect may be neutralised by a third force acting along their bisector in the opposite direction with a magnitude of (a) \(P\) (b) \(\frac{P}{2}\) (c) \(\frac{P}{\sqrt{2}}\) (d) \(\sqrt{2} P\)

The vector which can give unit vector along \(\mathrm{x}\)-axis with \(\quad \mathbf{A}=2 \hat{\mathbf{i}}-4 \hat{\mathbf{j}}+7 \hat{\mathbf{k}}, \quad \mathbf{B}=7 \hat{\mathbf{i}}+2 \hat{\mathbf{j}}-5 \mathbf{k} \quad\) and \(\mathbf{C}=-4 \hat{\mathbf{i}}+7 \hat{\mathbf{j}}+3 \hat{\mathbf{k}}\) is (a) \(4 \hat{\mathrm{i}}+5 \hat{\mathrm{j}}+5 \hat{\mathrm{k}}\) (b) \(-5 \hat{i}-5 \hat{j}+5 \hat{k}\) (c) \(-4 \hat{\mathrm{i}}-5 \hat{\mathrm{j}}-5 \hat{\mathrm{k}}\) (d) \(4 \hat{i}-5 \hat{j}-5 \hat{k}\)

What is the numerical value of the vector \(3 \hat{\mathbf{i}}+4 \hat{\mathbf{j}}+4 \hat{\mathbf{k}}\) ? (a) \(3 \sqrt{2}\) (b) \(5 \sqrt{2}\) (c) \(7 \sqrt{2}\) (d) \(9 \sqrt{2}\)

\(\mathbf{A}=3 \hat{\mathbf{i}}-\hat{\mathbf{j}}+7 \mathbf{k}\) and \(\mathbf{B}=5 \hat{\mathbf{i}}-\hat{\mathbf{j}}+9 \hat{\mathbf{k}}\). The direction cosine, \(m\) of the vector \(\mathbf{A}+\mathbf{B}\) is (a) zero (b) \(\frac{3}{\sqrt{31}}\) (c) \(\frac{8}{\sqrt{336}}\) (d) 5

If \(\mathbf{A}=2 \hat{\mathbf{i}}+3 \hat{\mathbf{j}}+6 \hat{\mathbf{k}}\) and \(\mathbf{B}=3 \hat{\mathbf{i}}-6 \hat{\mathbf{j}}+2 \hat{\mathbf{k}}\), then vector perpendicular to both \(\mathbf{A}\) and \(\mathbf{B}\) has magnitude \(k\) times that of \((6 \hat{\mathbf{i}}+2 \hat{\mathbf{j}}-2 \hat{\mathbf{k}})\). That \(k\) is equal to (a) 1 (b) 4 (c) 7 (d) 9

Given \(\mathbf{A}=\hat{\mathbf{i}}+2 \hat{\mathbf{j}}-3 \hat{\mathbf{k}}\). When a vector \(\mathbf{B}\) is added to \(\mathbf{A}\), we get a unit vector along X-axis. Then, \(\mathbf{B}\) is (a) \(-2 \hat{\mathbf{j}}+3 \hat{\mathrm{k}}\) (b) \(-\hat{\mathrm{i}}-2 \hat{\mathrm{j}}\) (c) \(-\hat{\mathbf{i}}+3 \hat{\mathrm{k}}\) (d) \(2 \hat{\mathrm{j}}-3 \hat{\mathrm{k}}\)

A proton of velocity \((3 \hat{\mathbf{i}}+2 \hat{\mathbf{j}}) \times 10^{5} \mathrm{~ms}^{-1}\) enters a magnetic field \((2 \hat{\mathbf{i}}+3 \hat{\mathbf{k}}) \mathrm{T}\). If the specific charge is \(9.6 \times 10^{7} \mathrm{C} \mathrm{kg}^{-1}\), the acceleration of the proton in \(\mathrm{ms}^{-2}\) is (a) \((6 \hat{i}-9 \hat{j}+4 \hat{k}) \times 9.6 \times 10^{12}\) (b) \((6 \hat{\mathrm{i}}+9 \hat{\mathrm{j}}+4 \hat{\mathrm{k}}) \times 9.6 \times 10^{12}\) (c) \((6 \hat{i}-9 \hat{j}-4 \hat{k}) \times 9.6 \times 10^{12}\) (d) \((6 \hat{i}+9 \hat{j}-4 \hat{k}) \times 9.6 \times 10^{12}\)

Two forces \(\mathbf{F}_{1}\) and \(\mathbf{F}_{2}\) are acting at right angles to each other. Then their resultant is (a) \(F_{i}+F_{2}\) (b) \(\sqrt{F_{1}^{2}+F_{2}^{2}}\) (c) \(\sqrt{F_{1}^{2}-E_{2}^{2}}\) (d) \(\frac{F_{1}+E_{2}}{2}\)

The \(x\) and \(y\) components of a force are \(2 \mathrm{~N}\) and \(-3 \mathrm{~N}\). The force is (a) \(2 \hat{\mathrm{i}}-3 \hat{\mathrm{j}}\) (b) \(2 \hat{\mathrm{i}}+3 \hat{\mathrm{j}}\) (c) \(-2 \hat{i}-3 \hat{j}\) (d) \(3 \hat{\mathrm{i}}+2 \hat{\mathrm{j}}\)

The sum of the magnitudes of two forces acting at a point is \(16 \mathrm{~N}\). The resultant of these forces is perpendicular to the smaller force has a magnitude of \(8 \mathrm{~N}\). If the smaller force is magnitude \(x\), then, the value of \(x\) is (a) \(2 \mathrm{~N}\) (b) \(4 \mathrm{~N}\) (c) \(6 \mathrm{~N}\) (d) \(7 \mathrm{~N}\)

Given \(\mathbf{R}=\mathbf{A}+\mathbf{B}\) and \(R=A=B .\) The angle between \(\mathbf{A}\) and \(\mathbf{B}\) is (a) \(60^{\circ}\) (b) \(90^{\circ}\) (c) \(120^{\circ}\) (d) \(180^{\circ}\)

The magnitude of the \(X\) and \(Y\) components of \(\mathbf{A}\) are 7 and 6. Also the magnitudes of \(X\) and \(Y\) components of \(\mathbf{A}+\mathbf{B}\) are 11 and 9 respectively. What is the magnitude of \(\mathbf{B}\) ? (a) 5 (b) 6 (c) 8 (d) 9

If \(\mathbf{A}_{1}\) and \(\mathbf{A}_{2}\) are two , \(2 \mathrm{~N}\) non- collinear unit vectors and if \(\left|\mathbf{A}_{1}+\mathbf{A}_{2}\right|=\sqrt{3}\) then the value of \(\left(\mathbf{A}_{1}-\mathbf{A}_{2}\right) \cdot\left(2 \mathbf{A}_{1}+\mathbf{A}_{2}\right)\) is (a) 1 (b) \(1 / 2\) (c) \(3 / 2\) (d) 2

One of the rectangular components of a velocity of \(60 \mathrm{kmh}^{-1}\) is \(30 \mathrm{kmh}^{-1}\). The other rectangular component is (a) \(30 \mathrm{kmh}^{-1}\) (b) \(30 \sqrt{3} \mathrm{kmh}^{-1}\) (c) \(30 \sqrt{2} \mathrm{kmh}^{-1}\) (d) zero

Two vectors a and \(\mathbf{b}\) are at an angle of \(60^{\circ}\) with each other. Their resultant makes an angle of \(45^{\circ}\) with a. If \(|\mathbf{b}|=2\) units, then \(|\mathbf{a}|\) is (a) \(\sqrt{3}\) (b) \(\sqrt{3}-1\) (c) \(\sqrt{3}+1\) (d) \(\sqrt{3} / 2\)

The angle between the z-axis and the vector \(\hat{\mathbf{i}}+\hat{\mathbf{j}}+\sqrt{2} \hat{\mathbf{k}}\) is (a) \(30^{\circ}\) (b) \(45^{\circ}\) (c) \(60^{\circ}\) (d) \(90^{\circ}\)

The \(x-y\) plane is the boundary between two transparent media. A medium I has a refractive index \(\mu_{1}=\sqrt{2}\) and medium II has a refractive index \(\mu_{2}=\sqrt{3}\). A ray of light in medium I, given by vector, \(\mathbf{A}=\sqrt{3} \hat{\mathbf{i}}-\hat{\mathbf{k}}\) is incident on the plane of separation. The unit vector in the direction of the refracted ray in medium II is (a) \(\frac{1}{\sqrt{2}}(\hat{\mathrm{i}}+\hat{\mathrm{k}})\) (b) \(\frac{1}{\sqrt{2}}(\hat{\mathrm{i}}+\hat{\mathrm{j}})\) (c) \(\frac{1}{\sqrt{2}}(\hat{\mathrm{k}}-\hat{\mathrm{i}})\) (d) \(\frac{1}{\sqrt{2}}(\hat{i}-\hat{k})\)

The resultant of two forces, each \(\mathbf{P}\), acting at an angle \(\theta\) is (a) \(2 P \sin \frac{\theta}{2}\) (b) \(2 P \cos \frac{\theta}{2}\) (c) \(2 P \cos \theta\) (d) \(P \sqrt{2}\)

The resultant of two vectors of magnitudes \(2 A\) and \(\sqrt{2} A\) acting at an angle \(\theta\) is \(\sqrt{10} A\). The correct value of \(\theta\) is (a) \(30^{\circ}\) (b) \(45^{\circ}\) (c) \(60^{\circ}\) (d) \(90^{\circ}\)

If, \(0.5 \hat{\mathbf{i}}+0.8 \hat{\mathbf{j}}+c \hat{\mathbf{k}}\) is a unit vector, then the value of \(c\) is (a) \(\sqrt{0.11}\) (b) \(\sqrt{0.22}\) (c) \(\sqrt{0.33}\) (d) \(\sqrt{0.89}\)

The vectors \(2 \hat{\mathbf{i}}+3 \hat{\mathbf{j}}-2 \hat{\mathbf{k}}, \quad 5 \hat{\mathbf{i}}+a \hat{\mathbf{j}}+\hat{\mathbf{k}} \quad\) and \(-\hat{\mathbf{i}}+2 \hat{\mathbf{j}}+3 \hat{\mathbf{k}}\) are coplanar when ' \(a\) ' is (a) \(-9\) (b) 9 (c) \(-18\) (d) 18

A vector \(\mathbf{A}\) when added to the vector \(\mathbf{B}=3 \hat{\mathbf{i}}+4 \hat{\mathbf{j}}\) yields a resultant vector that is in the positive \(y\)-direction and has a magnitude equal to that of \(\mathbf{B}\). Find the magnitude of \(\mathbf{A}\) (a) \(\sqrt{10}\) (b) 10 (c) 5 (d) \(\sqrt{15}\)

If \(\mathbf{P}=4 \hat{\mathbf{i}}-2 \hat{\mathbf{j}}+6 \hat{\mathbf{k}}\) and \(\mathbf{Q}=\hat{\mathbf{i}}-2 \hat{\mathbf{j}}-3 \hat{\mathbf{k}}\), then the angle which \(\mathbf{P}+\mathbf{Q}\) makes with \(x\)-axis is (a) \(\cos ^{-1}\left(\frac{3}{\sqrt{50}}\right)\) (b) \(\cos ^{-1}\left(\frac{4}{\sqrt{50}}\right)\) (c) \(\cos ^{-1}\left(\frac{5}{\sqrt{50}}\right)\) (d) \(\cos ^{-1}\left(\frac{12}{\sqrt{50}}\right)\)

In a two dimensional motion of a particle, the particle moves from point \(A\), position vector \(\mathbf{r}_{1}\) to point \(B\), position vector \(\mathbf{r}_{2}\). If the magnitudes of these vectors are respectively, \(r_{1}=3\) and \(r_{2}=4\) and the angles they make with the \(x\)-axis are \(\theta_{1}=75^{\circ}\) and \(15^{\circ}\), respectively, then find the magnitude of the displacement vector (a) 15 (b) \(\sqrt{13}\) (c) 17 (d) \(\sqrt{15}\)

If \(\mathbf{A}+\mathbf{B}=\mathbf{C}\) and \(A=\sqrt{3}, B=\sqrt{3}\) and \(C=3\), then the angle between \(\mathbf{A}\) and \(\mathbf{B}\) is (a) \(0^{\circ}\) (b) \(30^{\circ}\) (c) \(60^{\circ}\) (d) \(90^{\circ}\)

The resultant of two vectors \(\mathbf{A}\) and \(\mathbf{B}\) is perpendicular to the vector \(\mathbf{A}\) and its magnitude is equal to half of the magnitude of vector \(\mathbf{B}\). Then, the angle between \(\mathbf{A}\) and \(\mathbf{B}\) is (a) \(30^{\circ}\) (b) \(45^{\circ}\) (c) \(150^{\circ}\) (d) \(120^{\circ}\)

The angle between \(\mathbf{A}=\hat{\mathbf{i}}+\hat{\mathbf{j}}\) and \(\mathbf{B}=\hat{\mathbf{i}}-\hat{\mathbf{j}}\) is (a) \(45^{\circ}\) (b) \(90^{\circ}\) (c) \(-45^{\circ}\) (d) \(180^{\circ}\)

The magnitude of resultant of three vectors of magnitude 1,2 and 3 whose directions are those of the sides of an equilateral triangle taken in order is (a) zero (b) \(2 \sqrt{2}\) unit (c) \(4 \sqrt{3}\) unit (d) \(\sqrt{3}\) unit

If the magnitude of the sum of the two vectors is equal to the difference of their magnitudes, then the angle between vectors is (a) \(0^{\circ}\) (b) \(45^{\circ}\) (c) \(90^{\circ}\) (d) \(180^{\circ}\)

The area of the parallelogram represented by the vectors, \(\mathbf{A}=4 \hat{\mathbf{i}}+3 \hat{\mathbf{j}}\) and \(\mathbf{B}=2 \hat{\mathbf{i}}+4 \hat{\mathbf{j}}\) is (a) 14 units (b) \(7.5\) units (c) 10 units (d) 5 units

The simple sum of two co-initial vectors is 16 units. Their vector sum is 8 units. The resultant of the vectors is perpendicular to the smaller vector. The magnitudes of the two vectors are (a) 2 units and 14 units (b) 4 units and 12 units (c) 6 units and 10 units (d) 8 units and 8 units

If, the resultant of two forces \((A+B)\) and \((A-B)\) is \(\sqrt{A^{2}+B^{2}}\), then the angle between these forces is (a) \(\cos ^{-1}\left[-\frac{\left(A^{2}-B^{2}\right)}{A^{2}+B^{2}}\right]\) (b) \(\cos ^{-1}\left[-\frac{\left(A^{2}+B^{2}\right)}{\left(A^{2}-B^{2}\right)}\right]\) (c) \(\cos ^{-1}\left[-\frac{A^{2}+B^{2}}{2\left(A^{2}-B^{2}\right)}\right]\) (d) \(\cos ^{-1}\left[-\frac{2\left(A^{2}+B^{2}\right)}{A^{2}-B^{2}}\right]\)

If, \(\mathbf{A}=\hat{\mathbf{i}}+\hat{\mathbf{j}}-2 \hat{\mathbf{k}} \quad\) and \(\mathbf{B}=2 \hat{\mathbf{i}}-\hat{\mathbf{j}}+\hat{\mathbf{k}}\), then the magnitude of \(2 A-3 B\) is (a) \(\sqrt{90}\) (b) \(\sqrt{50}\) (c) \(\sqrt{190}\) (d) \(\sqrt{30}\)

Angle between \(\mathbf{A}\) and \(\mathbf{B}\) is \(\theta\). What is the value of \(\mathbf{A} \cdot(\mathbf{B} \times \mathbf{A}) ?\) (a) \(A^{2} B \cos \theta\) (b) \(A^{2} B \sin \theta \cos \theta\) (c) \(A^{2} B \sin \theta\) (d) zero

If the resultant of \(\mathbf{A}\) and \(\mathbf{B}\) makes angle \(\alpha\) with \(\mathbf{A}\) and \(\beta\) with \(\mathbf{B}\), then (a) \(\alpha<\beta\), always (b) \(\alpha<\beta\), if \(AB\) (d) \(\alpha<\beta\), if \(A=B\)

A body constrained to move in \(y\)-direction, is subjected to a force given by \(\mathbf{F}=(-2 \hat{\mathbf{i}}+15 \hat{\mathbf{j}}+6 \hat{\mathbf{k}})\) done by this force in moving the body through a distance of \(10 \mathrm{~m}\) along \(y\)-axis? (a) \(190 \mathrm{~J}\) (b) \(160 \mathrm{~J}\) (c) \(150 \mathrm{~J}\) (d) \(20 \mathrm{~J}\)

A proton in a cyclotron changes its velocity from \(30 \mathrm{kms}^{-1}\) north to \(40 \mathrm{kms}^{-1}\) east in \(20 \mathrm{~s}\). What is the average acceleration during this time (a) \(2.5 \mathrm{~km} \mathrm{~s}^{-2}\) at \(37^{\circ} \mathrm{E}\) of \(\mathrm{S}\) (b) \(2.5 \mathrm{~km} \mathrm{~s}^{-2}\) at \(37^{\circ} \mathrm{N}\) of \(\mathrm{E}\) (c) \(2.5 \mathrm{~km} \mathrm{~s}^{-2}\) at \(37^{\circ} \mathrm{N}\) of \(\mathrm{S}\) (d) \(2.5 \mathrm{~km} \mathrm{~s}^{-2}\) at \(37^{\circ} \mathrm{E}\) of \(\mathrm{N}\)

Consider a vector \(\mathbf{F}=4 \hat{\mathbf{i}}-3 \hat{\mathbf{j}}\). Another vector that is perpendicular to \(\mathbf{F}\) is (a) \(4 \hat{\mathrm{i}}+3 \hat{\mathrm{j}}\) (b) \(6 \hat{\mathrm{j}}\) (c) \(7 \hat{j}\) (d) \(3 \hat{\mathrm{i}}-4 \hat{\mathrm{j}}\)

Work done when a force, \(\mathbf{F}=(\hat{\mathbf{i}}+2 \hat{\mathbf{j}}+3 \hat{\mathbf{k}}) \mathbf{N}\) acting on a particle takes it from the point \(\mathbf{r}_{1}=(\hat{\mathbf{i}}-\hat{\mathbf{j}}+\hat{\mathbf{k}}) \mathrm{m}\) to the point \(\mathbf{r}_{2}=(\hat{\mathbf{i}}+\hat{\mathbf{j}}+2 \hat{\mathbf{k}}) \mathrm{m}\) is (a) \(-3 \mathrm{~J}\) (b) \(-1 \mathrm{~J}\) (c) zero (d) \(2 \mathrm{~J}\)

The resultant of two forces at right angle is \(5 \mathrm{~N}\). When the angle between them is \(120^{\circ}\), the resultant is \(\sqrt{13}\). Then, the forces are (a) \(\sqrt{12} \mathrm{~N}, \sqrt{13} \mathrm{~N}\) (b) \(\sqrt{20} \mathrm{~N}, \sqrt{5} \mathrm{~N}\) (c) \(3 \mathrm{~N}, 4 \mathrm{~N}\) (d) \(\sqrt{40} \mathrm{~N}, \sqrt{15} \mathrm{~N}\)

The radius vector and linear momentum are respectively given by vector \(2 \hat{\mathbf{i}}+2 \hat{\mathbf{j}}+\hat{\mathbf{k}}\) and \(2 \hat{\mathbf{i}}-2 \hat{\mathbf{j}}+\hat{\mathbf{k}}\). Their angular momentum is (a) \(2 \hat{\mathrm{i}}-4 \hat{\mathrm{j}}\) (b) \(4 \hat{i}-8 \hat{k}\) (c) \(2 \hat{\mathrm{i}}-4 \hat{\mathrm{j}}+2 \hat{\mathrm{k}}\) (d) \(4 \hat{i}-8 \hat{j}\)

If the resultant of the vectors \((\hat{\mathbf{i}}+2 \hat{\mathbf{j}}-\hat{\mathbf{k}}),(\hat{\mathbf{i}}-\hat{\mathbf{j}}+2 \hat{\mathbf{k}})\) and \(\mathbf{C}\) is a unit vector along the \(\mathrm{y}\)-direction, then \(\mathbf{C}\) is (a) \(-2 \hat{\mathrm{i}}-\hat{\mathrm{k}}\) (b) \(-2 \hat{\mathrm{i}}+\hat{\mathrm{k}}\) (c) \(2 \hat{\mathbf{i}}-\hat{\mathbf{k}}\) (d) \(-2 \hat{\mathrm{i}}+\hat{\mathrm{k}}\)

Three vectors \(\mathbf{A}, \mathbf{B}\) and \(\mathbf{C}\) add up to zero. Find which is false (a) \((A \times B) \times C\) is not zero unless \(B, C\) are parallel (b) \((A \times B) \cdot C\) is not zero unless \(B, C\) are parallel (c) If \(A, B, C\) defined a plane, \((A \times B) \times C\) is in that plane (d) \((\mathrm{A} \times \mathrm{B}) \cdot \mathrm{C}=|\mathrm{A}||\mathrm{B}||\mathrm{C}| \rightarrow C^{2}=A^{2}+B^{2}\)

Which one of the following statements is true? (a) A scalar quantity is the one that is conserved in a process (b) A scalar quantity is the one that can never take negative values (c) A scalar quantity is the one that does not vary from one point to another in space (d) A scalar quantity has the same value for observers with different orientations of the axes

If vectors \(\mathbf{A}\) and \(\mathbf{B}\) are given by \(\mathbf{A}=5 \hat{\mathbf{i}}+6 \hat{\mathbf{j}}+3 \mathbf{k}\) and \(\mathbf{B}=6 \hat{\mathbf{i}}-2 \hat{\mathbf{j}}-6 \hat{\mathbf{k}}\). Which is/are of the following correct? (a) A and B are mutually perpendicular (b) Product of \(A \times B\) is the same \(B \times A\) (c) The magnitude of \(\mathrm{A}\) and \(\mathrm{B}\) are equal (d) The magnitude of \(A \cdot B\) is zero

\((\mathbf{P}+\mathbf{Q})\) is a unit vector along \(X\)-axis. If, \(\mathbf{P}=\hat{\mathbf{i}}-\hat{\mathbf{j}}+\hat{\mathbf{k}}\) then what value is \(\mathbf{Q}\) ? (a) \(\hat{\mathrm{i}}+\hat{\mathrm{j}}-\hat{\mathrm{k}}\) (b) \(\hat{\mathrm{j}}-\hat{\mathrm{k}}\) (c) \(\hat{i}+\hat{j}+\hat{k}\) (d) \(\hat{\mathrm{j}}+\hat{\mathbf{k}}\)

It is found that \(|\mathbf{A}+\mathbf{B}|=|\mathbf{A}|\). This necessarily implies, (a) \(\mathrm{B}=0\) (b) A, B are antiparallel (c) \(\mathrm{A}, \mathrm{B}\) are perpendicular (d) \(\mathrm{A} \cdot \mathrm{B} \leq 0\)

What vector must be added to the sum of two vectors \(2 \hat{\mathbf{i}}-\hat{\mathbf{j}}+3 \hat{\mathbf{k}}\) and \(3 \hat{\mathbf{i}}-2 \hat{\mathbf{j}}-2 \hat{\mathbf{k}}\) so that the resultant is a unit vector along \(Z\)-axis? (a) \(5 \hat{\mathbf{i}}+\hat{\mathrm{k}}\) (b) \(-5 \hat{i}+3 \hat{j}\) (c) \(3 \hat{j}+5 \hat{k}\) (d) \(-3 \hat{j}+2 \hat{\mathrm{k}}\)

Which of the following statements is/are correct? (a) The magnitude of the vector \(3 \hat{i}+4 \hat{j}\) is 5 (b) A force \((3 \hat{i}+4 \hat{j}) \mathrm{N}\) acting on a particle causes a displacement \(6 \hat{j}\). The work done by the force is \(30 \mathrm{~N}\) (c) If \(\mathrm{A}\) and \(\mathrm{B}\) represent two adjacent sides of \(\mathrm{a}\) parallelogram, then \(|\mathrm{A} \times \mathrm{B}|\) give the area of that parallelogram (d) A force has magnitude \(20 \mathrm{~N}\). Its component in a direction making an angle \(60^{\circ}\) with the force is \(10 \sqrt{3} \mathrm{~N}\)

For two vectors \(\mathbf{A}\) and \(\mathbf{B},|\mathbf{A}+\mathbf{B}|=|\mathbf{A}-\mathbf{B}|\) is always true when (a) \(|A|=|B| \neq 0\) (b) \(\mathrm{A} \perp \mathrm{B}\) (c) \(|\mathrm{A}|=|\mathrm{B}| \neq 0\) and \(\mathrm{A}\) and \(\mathrm{B}\) are parallel or anti parallel (d) when either |A|or |B| is zero.