Short FAQ - Unit 2 Scalars & Vectors

By Dr. Purushottam Bhandari, Liverpool College, New Baneshwor 1. Scalars and Vectors 1. Differentiate between scalars and vectors with examples. The following are some differences between scalars and vectors. Scalars Vectors The1 physical quantities that can be 1 The physical quantities that require both represented by magnitude alone are magnitude and direction to be represented called scalars. are called vectors. The2y can be represented as a number 2 They are represented by a number with or a letter alone to express their direction or by a bold-faced letter or by a magnitude. letter with arrow head. The3y follow the simple laws of 3 They follow the vector law of addition, addition, multiplication and division multiplication and division (e.g. of algebra. parallelogram law, triangle law, dot product, cross product etc.). The4y change with change in their 4 They change with change in their magnitudes only. magnitude or direction or both. Exa5mples: mass, time, length, energy, 5 Examples: force, velocity, acceleration, temperature, current, work done etc. momentum, torque etc. 2. How is a vector quantity represented graphically and symbolically? Symbolically, a vector is represented by a bold-faced letter or by a letter with arrow head (e.g. A or ⃗A). Graphically, a vector quantity is expressed as a line segment with an arrow. The length of the line segment represents the magnitude of the vector quantity whereas the arrow head and orientation of the line segment gives its direction. For example, the figure is the graphical representation of two vectors A⃗ and ⃗B, where vector A⃗ is half of vector ⃗B in magnitude. The vector A⃗ is directed backwards while vector ⃗B is directed forward. ������ ���⃗��� Figure 1: Graphical representation of vector 3. Is a physical quantity having magnitude and direction necessarily a vector quantity? Explain. [071, 063] No. For a quantity to be a vector it must follow the vector laws of algebra. For example, current has both magnitude and direction but it is a scalar quantity as it follows the simple laws of addition, multiplication and division of algebra. 4. Does vector addition hold true for any two vectors? Explain. 12

No. The vector addition holds true for the same kind of physical quantities i.e., having the same dimensions. For example, although force and velocity both are vectors, they cannot be added. 5. Define unit vector and null vector. Unit vector: The vector having its magnitude equal to unity is called a unit vector. It is represented by a hat (or circumflex) over a small letter. In figure, î, ĵ, and k̂ represent unit vectors along x-, y- and z-coordinates respectively. Null vector: A vector with zero magnitude and arbitrary direction is called a null or zero vector. It is denoted by ⃗0. For example, the addition of two equal and opposite vectors or subtraction of two equal vectors or a multiplication of a vector by a null vector give a null vector. y ������ + ������ = ������ − ������ = 0 ������̂ ������̂ x O Figure 2: Null Vector and unit vector z ���̂��� 6. If a vector has zero magnitude, is it meaningful to call it a vector? [069] The addition of two equal and opposite vectors or subtraction of a vector from equal vector or a multiplication of a vector by a zero vector give a vector with zero magnitude and is called a null vector. It is, therefore, a vector of zero magnitude can be a vector. 7. Can a vector be multiplied with both dimensional and nondimensional scalars? Yes. A vector can be multiplied with both dimensional and nondimensional scalars. When a vector is multiplied by a dimensional scalar, it results in different kind of vector with different dimension. For example, when a vector acceleration is multiplied by a dimensional scalar mass, the result is the vector force (F⃗ = m⃗a). On the other hand, when a vector (e.g. normal reaction) is multiplied by a nondimensional scalar (e.g. coefficient of friction), the resulting vector (e.g. friction force) is still a vector with the same dimension (⃗F = μ⃗R). 8. What are equal vectors? What is the negative of a vector? Two vectors are said to be equal vectors if they have the same magnitude and same direction. The negative of a vector is the vector having the same magnitude but just opposite direction. For example, ⃗A and A⃗ are equal vectors while −⃗A is a negative vector of A⃗ . 9. State and explain the parallelogram law of vector addition. [075, 070, 069, 068, 055] If two vectors acting simultaneously at a point can be represented both in magnitude and direction by the adjacent sides of a parallelogram drawn from a point, then the resultant vector is represented both in magnitude and direction by the diagonal of the parallelogram passing through that point. This is called the parallelogram law of vectors. ���⃗��� If vectors A⃗ and ⃗B are inclined at an angle of θ, then their resultant is given by ⃗R = ⃗A + B⃗ . The magnitude and direction are calculated as, R = √A2 + B2 + 2AB cos θ. ������ If ϕ be the angle made by resultant with vector ⃗A, then ������ ϕ = tan−1 ( ������ sin θ ) with A⃗ . ������ A+B cos θ Figure 3: Addition of vectors by parallelogram law 13

10. State and explain triangle law of vector addition. [066] Triangle law of vector addition states that when two vectors acting on a body are represented in magnitude and direction by two sides of a triangle taken in same order then the third side of that triangle taken in opposite order represents the magnitude and direction of their resultant. If A⃗ and ⃗B are two vectors acting along the two sides of a triangle, then their resultant ⃗R is given by as shown in the figure. ���⃗��� −���⃗��� ������ −������ Figure 4: Addition of vectors by triangle law 11. Explain how a vector ���⃗⃗��� can be added to a vector ⃗���⃗���. What about the subtraction of ⃗���⃗��� from ���⃗⃗��� ? The addition of two vectors ⃗A and ⃗B is represented by ⃗R = A⃗ + ⃗B, where R⃗ is the sum of the vectors and is also called the resultant of the given two vectors. The resultant is calculated using either a parallelogram law of vectors or using triangle law of vectors. The subtraction of vector ⃗B from vector ⃗A is similar to the addition of vector A⃗ and vector −⃗B i.e., ⃗R = ⃗A − ⃗B = A⃗ + (−⃗B). ���⃗��� ������ ������ −���⃗��� Figure 5: Addition and subtraction of vectors 12. What do you mean by composition and resolution of vectors? The process of finding the resultant vector of two or more than two vectors is called the composition of vectors. Parallelogram law and triangle laws are some examples of composition of vectors. The process of splitting a vector into two or more vectors in specified directions is called the resolution of vector. The vectors into which a given vector is split are called the component vectors. The component vectors acting together give the original vector as their resultant vector. 13. What do you mean by the rectangular components of a vector? When a vector is resolved along mutually perpendicular directions, the component vectors are called the rectangular components of the given vector. In two- dimension, there are two rectangular components. While in three- dimension, there are three. If A⃗ be a vector acting by making an angle θ with the x-axis as shown, then A⃗ x and ⃗Ay are the rectangular components of the vector ⃗A along x- and y- directions respectively. A⃗ = ⃗Ax + ⃗Ay = Axî + Ayĵ = A cos θ î + A sin θ j.̂ 14

yy ������������ = ������ sin ������ ������������ = ������ cos ������ ������ ������ ������ x ������ x ������������ = ������ cos ������ ������������ = ������ sin ������ Figure 6: Resolution of vectors into rectangular components 14. What do you mean by dot or scalar product of two vectors? If the product of two vectors is a scalar, such operation is called dot product. If A⃗ and ⃗B are two vectors and θ be the angle between them, then their dot product is given by ⃗A. ⃗B = AB cos θ. Thus, it is the product of magnitude of one vector and the magnitude of the component (or projection) of the other vector along the direction of the first vector. The dot product is important to find the angle between two vectors. It is used to see the effect of a vector along a specified direction. For example, if a force acts on a body at an angle θ along the direction of displacement of the body, then the work done by the force on displacing the body by a distance d is given by W = ⃗F. d⃗ = Fd cos θ. ������ ������ ������ cos ������ ������ Figure 7: Example of scalar product (work done) 15. What do you mean by vector or cross product of two vectors? If the product of two vectors is a vector, such operation is called cross product. If A⃗ and ⃗B are two vectors and θ be the angle between them, then their vector or cross product is given by A⃗ × ⃗B = AB sin θ n̂. Here, AB sin θ gives the area of a parallelogram that has the vectors as its sides. And, n̂ is a unit vector perpendicular to the plane containing ⃗A and ⃗B whose direction is given by right-hand rule or cork-screw rule. The cross product is used to observe the curling or turning or rotational effect of vectors. It can also be used to find the angle between two vectors and to find the area of parallelogram formed by the vectors. ������ × ���⃗��� ���⃗��� ������ ������ Figure 8: Example of cross product Note: Right-hand rule: When fingers curl in the direction from ⃗A to ⃗B, then extended thumb points in the direction of n̂ or A⃗ × ⃗B. Cork-screw rule: n̂ or A⃗ × ⃗B points along the direction of displacement of a screw if it is rotated from A⃗ to ⃗B. 16. Show that the resultant of two equal vectors divide the angle between the vectors equally. 15

From the relation, ϕ = tan−1 ( B sin θ θ). ������ A+B cos ������ ( A sin θ ) ������ If A⃗ = B⃗ , then ϕ = tan−1 A+A cos θ ������ = tan−1 ( sin θ ) = tan−1 (tan θ2). 1+cos θ ∴ ϕ = 2θ. Figure 9: Addition of two equal vectors 17. Distinguish between dot product and cross product of two vectors. [062] Dot Product Cross Product 1 If the product of two vectors 1 If the product of two vectors is is a scalar, such operation is a vector, such operation is called dot product. called cross product. 2 If ������ and ���⃗��� are two vectors and 2 If ������ and ���⃗��� are two vectors and ������ be the angle between ������ be the angle between them, them, then their dot product then their dot product is is given by ������. ���⃗��� = given by ������ × ���⃗��� = ������������ ������������������ ������. ������������ ������������������ ������ ���̂���. 3 It is commutative i.e., ������. ���⃗��� = 3 It is not commutative i.e., ������ × ���⃗��� ≠ ���⃗��� × ������. ���⃗���. ������. 4 The dot product of two 4 The cross product of two mutually perpendicular mutually perpendicular vectors is minimum (null or vectors is maximum i.e., ������ × ���⃗��� = ������������ ������������������ 90 ���̂��� = ���������������̂���. zero) i.e., ������. ���⃗��� = ������������ ������������������ 90 = 0. 5 The dot product of two 5 The cross product of two parallel vectors is maximum parallel vectors is null (zero) i.e., ������. ���⃗��� = ������������ ������������������ 0 = ������������. i.e., ������ × ���⃗��� = ������������ ������������������ 0 ���̂��� = 0���̂��� 6 The dot product of a vector 6 The cross product of a vector with itself is equal to the with itself is zero i.e., ������ × square of its magnitude i.e., ������ = ������������ ������������������ 0 ���̂��� = 0���̂���. ������. ������ = ������������ ������������������ 0 = ������2. 7 It is used to find the angle 7 It is used to find the curling or between two vectors or to turning or rotational effect, find the effect of one vector angle between the vectors onto another vector. and area of parallelogram formed by the vectors. 8 Example: Work done (������ = 8 Example: Torque (������ = ������ × ������). ������. ������). 16

18. If ���⃗⃗���. ⃗���⃗��� = ������, what is the angle between ���⃗⃗��� and ⃗���⃗���? [074] From A⃗ . ⃗B = AB cos θ, if ⃗A. ⃗B = 0, then AB cos θ = 0. It implies that either A = 0 or B = 0 or cos θ = 0. If A⃗ and ⃗B are non-zero vectors, then cos θ = 0 or θ = 90° i.e., the two vectors A⃗ and ⃗B are perpendicular to each other. 19. What does ���⃗⃗���. ���⃗⃗���, the scalar product of a vector with itself give? What about ���⃗⃗��� × ���⃗⃗���, the vector product of a vector with itself? [070] The dot or scalar product of a vector with itself gives the square of its magnitude. It is because the projection of a vector on to itself leaves its magnitude unchanged as the angle of a vector with itself is zero (A⃗ . A⃗ = AA cos θ = A2 cos 0 = A2). The cross or vector product of a vector with itself gives a zero or null vector since A⃗ × A⃗ = AA sin θ n̂ = A2 sin 0 n̂ = 0. The physical meaning of this is that the area of a parallelogram with only one side (or one side A and another side zero) is zero. 20. If ���⃗⃗��� is added to ���⃗⃗���, under what condition does the resultant vector have a magnitude equal to A + B? Under what conditions is the resultant vector equal to zero? [054] Or, ������ is the vector sum of ���⃗⃗��� and ⃗���⃗��� i.e. ������ = ���⃗⃗��� + ⃗���⃗��� for ������ = ������ + ������ to be true. What is the angle between ���⃗⃗��� and ���⃗⃗���? [067] If ⃗A and ⃗B are two vectors inclined at an angle θ and R⃗ is their resultant vector, then ⃗R = A⃗ + ⃗B, and R = √A2 + B2 + 2AB cos θ. Alternatively using dot product: ∴ R2 = A2 + B2 + 2AB cos θ. ���⃗���. ���⃗��� = ൫������ + ���⃗���൯. (������ + ���⃗���) For R = A + B, ������2 = ������. ������ + ������. ���⃗��� + ���⃗���. ������ + ���⃗���. ���⃗��� Or, A2 + B2 + 2AB = A2 + B2 + 2AB cos θ ������2 = ������2 + 2������������ cos ������ + ������2 Or, cos θ = 1 i. e. , θ = 00 Therefore, the resultant vector of addition of two vectors is equal to the sum of their magnitudes if the angle between them is 00 i.e., if they act along the same direction. For R = 0, A2 + B2 + 2AB cos θ = 0 If we assume ⃗A = ⃗B, then cos θ = 1 i. e. , θ = 1800. Therefore, the resultant vector of addition of two vectors is equal to zero if the two vectors are equal and angle between them is 1800 i.e., if equal vectors act along the opposite direction. 21. The magnitude of the resultant of two vectors ���⃗⃗��� and ⃗���⃗��� is given by ������������ = ������������ + ������������. What is the angle between ���⃗⃗��� and ⃗���⃗���? If ⃗A and ⃗B are two vectors inclined at an angle θ and ⃗R is their resultant vector, then R = √A2 + B2 + 2AB cos θ. If R2 = A2 + B2, then A2 + B2 = A2 + B2 + 2AB cos θ. Or, cos θ = 0 i. e. , θ = 900. Therefore, in this case the two vectors are inclined at an angle of 900. 22. Two vectors ���⃗⃗��� and ⃗���⃗��� are such that ������ = ���⃗⃗��� − ⃗���⃗��� and ������ = ������ − ������. Find the angle between them? [064] If A⃗ and ⃗B are two vectors inclined at an angle θ and if C⃗ = ⃗A − ⃗B and C = A − B, then C2 = A2 + B2 + 2AB cos(180 − θ). (A − B)2 = A2 + B2 + 2AB cos(180 − θ) 17

Or, cos(180 − θ) = −1 = cos 180 Alternatively using dot product: i. e. , θ = 00. ������. ������ = ൫������ − ���⃗���൯. (������ − ���⃗���) ������2 = ������. ������ − ������. ���⃗��� − ���⃗���. ������ + ���⃗���. ���⃗��� (������ − ������)2 = ������2 − 2������������ cos ������ + ������2 Or, cos ������ = 1 = cos 0 ������. ������. , ������ = 00. 23. Can the sum of the equal vectors be equal to either vector? [072, 061] Or, Resultant of two equal forces may have the magnitude equal to one of the forces. At what angle between the two equal forces this is possible? [074] Or, Two vectors have equal magnitudes and their resultant also has the same magnitude. What is the angle between the two vectors? [073, 072, 067, 061] If A⃗ and ⃗B are two vectors inclined at an angle θ and R⃗ is their resultant vector, and if A = B and R = A = B, then from R⃗ = A⃗ + ⃗B, and R = √A2 + B2 + 2AB cos θ, A = √A2 + A2 + 2AA cos θ or, A2 = A2 + A2 + 2AA cos θ or, cos θ = − 1 i. e. , θ = 1200 2 Therefore, for the sum of two vectors to be equal to either of them, the two vectors should incline at an angle of 1200. 24. Can the resultant of two unequal vectors be zero? What about of three vectors? If ⃗A and ⃗B are two vectors inclined at an angle θ and ⃗R is their resultant vector, then R = √A2 + B2 + 2AB cos θ . Here, R = A + B (maximum value) when θ = 00 R = A − B (minimum value if A ≠ B) when θ = 1800. Therefore, the resultant of two unequal vectors cannot be zero. However, the resultant of three vectors can be zero if all vectors lie in the same plane and the vector sum of any two of them is equal and opposite to the third vector, i.e., if ⃗A = −(⃗B + C⃗ ). 25. Can three vectors give zero resultant if (i) they lie in a plane (ii) they do not lie in a plane? (i) If three vectors lie in a same plane and if the vector sum of any two of them is equal and opposite to the third vector (i.e., if ⃗A = −(⃗B + ⃗C), then the resultant of them is zero. (ii) If all three vectors do not lie in a same plane then the resultant of them cannot be zero. It is because, any two vectors out of three must lie in a plane. Then, the resultant vector of these two vectors must also lie in the same plane containing the two vectors. For the net resultant to be zero, the resultant of the two vectors must be equal and opposite to the third vector, which is possible only if they lie in the same plane. 26. Can the resultant magnitude of two vectors be smaller than the magnitude of either vector? If A⃗ and ⃗B are two vectors inclined at an angle θ and ⃗R is their resultant vector, then R = √A2 + B2 + 2AB cos θ. If the value of θ is such that 900 < θ < 2700, then R can be smaller than either of A or B depending upon their values. For example, at θ = 1800, R = A − B. 27. If the sum and difference of two vectors are perpendicular to each other, prove that the vectors have equal magnitude. 18

Here, if ⃗A and ⃗B are two vectors, then according to question, (A⃗ + ⃗B) ⊥ (⃗A − ⃗B) i. e., ൫A⃗ + ⃗B൯. ൫A⃗ − ⃗B൯ = 0. or, ⃗A. ⃗A − ⃗A. ⃗B + ⃗B. ⃗A − ⃗B. ⃗B = 0 or, A2 − B2 = 0 or, A = ±B. Hence the magnitude of two vectors are equal. 28. If the sum and difference of two vectors are equal to each other, prove that the vectors are perpendicular to each other. If ⃗A and ⃗B are two vectors inclined at an angle θ, then according to question, |A⃗ + ⃗B| = |A⃗ − ⃗B| Squaring on both sides, ൫A⃗ + ⃗B൯. ൫⃗A + ⃗B൯ = ൫⃗A − ⃗B൯. (⃗A − ⃗B) or, ⃗A. A⃗ + A⃗ . ⃗B + ⃗B. ⃗A + ⃗B. ⃗B = A⃗ . ⃗A − A⃗ . ⃗B − ⃗B. A⃗ + ⃗B. ⃗B or, A2 + 2⃗A. ⃗B + B2 = A2 − 2⃗A. ⃗B + B2 (dot product is commutative) or, 4A⃗ . ⃗B = 0 or, AB cos θ = 0 or, cos θ = 0 i. e. θ = 900. 29. Can you find a vector quantity that has a magnitude of zero but components that are different from zero? Explain. [070] Let A⃗ be a vector and A⃗ x, A⃗ y and A⃗ z be its components along x-, y-, and z-axes respectively, then, A⃗ . A⃗ = A2 = A2x + A2y + A2z. According to question, if A = 0, then A2x + A2y + A2z = 0. This is possible only when Ax = Ay = Az = 0 as the sum of the square of nonzero quantities cannot be zero. Therefore, it is not possible to have a vector quantity that has a magnitude of zero but components that are different from zero. 30. Can a vector be zero when one of its components is not zero while all the remaining components are zero? Let ⃗A be a vector and A⃗ x, ⃗Ay and A⃗ z be its components along x-, y-, and z-axes respectively, then, ⃗A. ⃗A = A2 = A2x + A2y + A2z. According to question, if any two of the components, say, Ax = Ay = 0, then A2 = A2z. Now, for A⃗ = 0, it is also required that Az = 0. Therefore, it is not possible for a vector to be zero when one of its components is not zero while all the remaining components are zero. 31. Given two vectors ���⃗⃗��� = ������. ������������������̂ + ������. ������������������̂ and ���⃗⃗��� = ������. ������������������̂ − ������. ������������������̂. Find the magnitude of each vector. [072] Here, A⃗ = 4.00î + 3.00ĵ and ⃗B = 5.00î − 2.00ĵ ∴ A⃗ . A⃗ = A2 = A2x + A2y + A2z = 16 + 9 + 0 = 25 ∴ A = 5.00. Similarly, ⃗B. ⃗B = B2 = 25 + 4 = 29 ∴ B = √29. 32. If ���⃗⃗��� = ������������̂ − ������̂ + ���������̂��� and ⃗���⃗��� = ������������̂ + ������������̂ + ���̂���, find the angle between the vectors ���⃗⃗��� and ⃗���⃗���? [076] If θ be the angle between the two given vectors, then their scalar product is given by ⃗A. ⃗B = AB cos θ. Then, cos θ = A⃗⃗ .⃗B⃗ |A||B| ⃗A. ⃗B = ൫4î − ĵ + 3k̂൯. ൫7î + 5ĵ + k̂൯ = (28) + (−5) + (3) = 26 19

⃗A. ⃗A = A2 = A2x + A2y + A2z = 16 + 1 + 9 = 26 ⃗B. ⃗B = B2 = Bx2 + By2 + Bz2 = 49 + 25 + 1 = 75 Therefore, cos θ = A⃗⃗ .⃗B⃗ = 26 ⇒ θ = cos−1(√26) = 54°. |A||B| √26 √75 75 33. A vector ������ = ������̂ + ������������̂ − ���������̂��� is given, what are the magnitude of the x-, y- and z- components of the vector? [072] Here, F⃗ = F⃗ x + ⃗Fy + ⃗Fz = Fxî + Fyĵ + Fkk̂ Therefore, comparing this equation with the given equation F⃗ = î + 2ĵ − 3k̂, The magnitude of x-component of F⃗ = Fx = 1 The magnitude of y-component of F⃗ = Fy = 2 The magnitude of z-component of ⃗F = Fz = −3. 34. A vector is defined as ���⃗��� = ������������̂ + ������������̂ − ���������̂���. What is the magnitude of y-component of ���⃗���? [073] The y-component of E⃗ is 3. (See Q.N. 33) 35. Two vectors are given as ���⃗⃗��� = ������������̂ + ������������̂ + ���������̂��� and ⃗���⃗��� = ������������̂ − ������������̂ − ���������̂���. Which one of the two is larger in magnitude? Justify your answer. [074] Given that, A⃗ = 2î + 3ĵ + 4k̂ and ⃗B = 3î − 2ĵ − 4k̂ Here, ⃗A. ⃗A = A2 = A2x + A2y + A2z = 4 + 9 + 16 = 39 ⃗B. ⃗B = B2 = Bx2 + By2 + Bz2 = 9 + 4 + 16 = 29 It is seen that the magnitude of ⃗A (which is √39) is greater than that of ⃗B (which is √29). 36. If the scalar product of two vectors is equal to the magnitude of their vector product, find the angle between them? [060] Or, ���⃗⃗��� and ���⃗⃗��� are two non-zero vectors. If |���⃗⃗��� × ���⃗⃗���| = ���⃗⃗���. ���⃗⃗���, what is the angle between ���⃗⃗��� and ⃗���⃗���? [075] According to question, if the scalar product of two vectors is equal to the magnitude of their vector product and if θ be the angle between any two vectors A⃗ and ⃗B, then A⃗ . ⃗B = ⃗A × ⃗B. or, AB cos θ = AB sin θ or, tan θ = 1 = tan 45 i. e. , θ = 450. 37. The magnitude of two vectors are 3 and 4 and their product is 6. What is the angle between them? [066] If ⃗A and ⃗B are two vectors inclined at an angle θ, then if the product of these vectors is 6, then ⃗A. ⃗B = AB cos θ = 6 But, given that A = 3 and B = 4, (3)(4) cos θ = 6 or, cos θ = 1 = cos 60 2 ∴ θ = 600. 38. If ������̂, ������̂ and ���̂��� are the unit vectors along x, y and z –axis respectively, find ������̂. (���̂��� × ������̂). [070] The given question is the scalar product of two vectors î and k̂ × j.̂ Therefore, first we need to find the cross product of k̂ × ĵ, which is 20

k̂ × ĵ = (1)(1) sin 90 (−i)̂ = −î Now, î. ൫k̂ × ĵ൯ = î. (−î) = (1)(1) cos 180 = −1. 39. ������̂, ������̂ and ���̂��� are the unit vectors along x, y and z –axis respectively. Find the magnitude and direction of the vector product of two forces ������������ and ������������ if ������������ = ������������̂ and ������������ = −���������̂���. [069] The cross (or vector) product of the given forces is given by ⃗F1 × F⃗ 2 = (3î) × ൫−2k̂൯ = −6൫î × k̂൯ = −6(−ĵ) = 6j.̂ Thus, the magnitude of cross product of ⃗F1 and F⃗ 2 is 6. Its direction is along the y-axis i.e., along perpendicular to the plane containing F⃗ 1 and F⃗ 2. 40. The angle between two vectors ���⃗⃗��� and ���⃗⃗��� is ������. What are the magnitude and direction of ���⃗⃗���. ⃗���⃗��� and ���⃗⃗��� × ⃗���⃗���? [070] The magnitude of ⃗A. ⃗B (which is a scalar product of is ⃗A and ⃗B) is AB cos θ. Since, the scalar product of two vectors is a scalar, it does not have any direction. The magnitude of ⃗A × ⃗B (which is a vector product of ⃗A and ⃗B) is AB sin θ and its direction is along the perpendicular to the plane containing ⃗A and ⃗B as given by corkscrew right-hand rule. 41. If ���⃗⃗��� and ⃗���⃗��� are non-zero vectors, is it possible for ���⃗⃗���. ⃗���⃗��� and ���⃗⃗��� × ���⃗⃗��� both to be zero? [069] It is not possible. If ⃗A and ⃗B are two vectors inclined at an angle θ, then for the scalar product to be zero, ⃗A. ⃗B = AB cos θ = 0 and if ⃗A ≠ 0 and ⃗B ≠ 0, then cos θ = 0 i. e. , θ = 900. Now, if θ = 900, then the cross product of ⃗A and ⃗B is given by, A⃗ × ⃗B = AB sin θ n̂ = AB sin 90 n̂ = AB(1)n̂ = ABn̂. Since, ⃗A ≠ 0 and ⃗B ≠ 0, then A⃗ × ⃗B ≠ 0 for θ = 900. Therefore, it is not possible for ⃗A. ⃗B and A⃗ × ⃗B both to be zero if ⃗A and ⃗B are non zero vectors. 42. A force (in Newton) expressed in vector form as ������ = ������������̂ + ������������̂ − ���������̂��� is applied on a body and produces a displacement (in meter) ⃗���⃗��� = ������������̂ − ������������̂ − ���������̂��� in 4 seconds. Estimate the work done and power. [071] Here, the work done and power are given by, W = F⃗ . ⃗D⃗ and P = W t Here, W = F⃗ . D⃗⃗ = ൫4î + 7ĵ − 3k̂൯. ൫3î − 2ĵ − 5k̂൯ = 12 − 14 + 15 = 13Joule. P = W = 13 = 3.25Watt. t 4 43. What is a torque of a force ������ = ������������̂ − ������������̂ + ���������̂��� N acting at a point ������ = ������������̂ + ������������̂ + ���������̂��� meter about the origin? Since torque is given by the cross product of force and radius vector, then τ⃗ = r × F⃗ = (3î + 2ĵ + 3k̂) × (2î − 3ĵ + 4k̂) = 3î × ൫2î − 3ĵ + 4k̂൯ + 2ĵ × ൫2î − 3ĵ + 4k̂൯ + 3k̂ × (2î − 3ĵ + 4k̂) = 6(î × î) − 9(î × ĵ) + 12൫î × k̂൯ + 4(ĵ × i)̂ − 6(ĵ × ĵ) + 8൫ĵ × k̂൯ + 6൫k̂ × î൯ − 9൫k̂ × j൯̂ + 12൫k̂ × k̂൯ = 0 − 9k̂ + 12(−ĵ) + 4൫−k̂൯ − 0 + 8(i)̂ + 6(ĵ) − 9(−î) + 0 = 17î + 6ĵ − 13k̂. 21

∴ ⃗τ = r × F⃗ = 17î + 6ĵ − 13k̂. 44. What will happen to the resultant, dot product and cross product if the directions of two vectors are reversed? If the vectors are reversed then the direction of their resultant vector is also reversed as shown in Figure 10. However, there is no any change in dot product and cross product of two vectors if the components of vectors are reversed. ���⃗��� ������ −������ ������ ������ −���⃗��� Figure 10: Addition of vectors 22