Chapter#01: Vector Analysis

Team Quanta gladly presents all possible short questions of BS Physics book Mechanics – I’s Chapter#01: Vector Analysis for the ease of students.

Q.1 Can a scalar product of two vectors be negative? If your answer is yes, give an example. If not, give proof.

Answer: Yes, if vectors are oriented at an angle of 180°.

Example; Work done by forces of constraints e.g. friction is negative.

Q.2 A plane is moving at 100km/h at an angle of 60° with ground. What is speed of its projection on ground if the sun is just above plane at noon?

Answer: It is given that projection of plane is moving on ground, say along x axis, then we have to find x-component of its velocity:

V_x =v =100km/hxcos60° = 50km/h

Q.3 Consider a vector A with components A_xand A_y < 0.Find possible directions of vector A .

Answer: The vector A lies in the third quadrant.

Q.4 Under what conditions would a vector have components equal in magnitude?

Answer : A vector has equal rectangular components if it is inclined at 45° to coordinate axes.

Q.5 Does the value of vector quantity depends upon selected coordinate system?

Answer: The value of vector quantity does not change with change of reference axes since direction of vector is specified by angle made with reference axis.

Q.6 Weather gradient of a scalar field depends upon any particular system of coordinates?

Answer: No, gradient of a scalar field is entirely independent of any particular system of coordinates. Such quantities which are independent of coordinate system are called invariants.

Q.7 Suppose C= A+ B, does it follow that either C ≥ A or C ≥ B? If not, why? If yes prove it.

Answer: Suppose $\vec{C} = \vec{A}+\vec{B}$ , then C= A if B=0 and C=B if A=B, otherwise C > A or C > B

Q.8 Suppose is non-zero vector. It is given that AXB=0 and . A.B=0. What can you conclude about B.

Answer: From vectors, we know that

$|\vec{A}\ x\ \vec{B}|2 + \left|\vec{A}.\vec{B}\right|2 = A2 B2$

Using given values,

0= A²B²→ AB = 0

Since A≠0, showing that B is a null vector.

Q.9 If all components of vectors and are reversed, how will this affect

Answer: By reversing vectors, we have

$\vec{A} \times \vec{B} = -\vec{A} \times \vec{B} = \vec{A} \times \vec{B}$

This shows that vector product is unaltered if vectors are reversed.

O.10 A vector lies in xy-plane. For what orientation will both rectangular components be negative? For what orientation will its components have opposite signs?

Answer: Both rectangular components of be negative when it lies in 3^rd quadrant.

Both rectangular components of be positive in sign when it lies in 2^nd and fourth quadrant.

Q.11 Scalar triple product is cyclic. Can you give such a statement for vector triple product?

Answer: No, vector triple product is not cyclic.

Q. 12 Can we say that cylindrical polar coordinates are orthogonal?

Answer: Yes, we can say that cylindrical polar coordinates are orthogonal.

Q.13 What is the physical significance of derivative?

Answer: Consider a curve given by equation,

y = f (x)

The derivation $\frac{dy}{dx}$ represents slop of tangent line to above curve.

Q.14 What is geometrical interpretation of gradient of a scalar field?

Answer: Gradient of a scalar field ‘S’ is a vector whose magnitude is equal to maximum rate of change of ‘S’ and its direction is same as direction of displacement along which rate of change is maximum.

When we move in a direction normal to direction of S, then

$\frac{dS}{dr} =0 \rightarrow S$ = constant. This defines that a unit vector normal to surface S= constant is given by,

$\hat{n} = \frac{\nabla\phi}{\left|\nabla\phi\right|}$

Q.15 You know that “ line integral of a conservative field is independent of path”. Prove this statement.

Answer: A conservative field is derivable from a scalar field .

Line integral of this conservative field is,

$\vec{A}= -\nabla \phi$

Line integral of this conservative field A is,

$\oint{\vec{A}.\vec{dr}} = - \oint{\nabla\phi.\vec{dr}}$

$\oint{\vec{A}.\vec{dr}} = - \oint\left[\frac{\partial\phi}{\partial x}\ \hat{i}+\ \frac{\partial\phi}{\partial y}\hat{j}+\frac{\partial\phi}{\partial z}\hat{k}\right]. \left(dx\hat{i}+dy\hat{j}+\ dz\hat{k}\right)$

$\oint{\vec{A}.\vec{dr}} = -\oint\left[\frac{\partial\phi}{\partial x}\ dx+\ \frac{\partial\phi}{\partial y}dy+\frac{\partial\phi}{\partial z}dz\right]$

$\oint{\vec{A}.\vec{dr}} = - \int_{A}^{B}d\phi = - (\phi A - \phi B )$

Above equation shows that line integral of a conservative field depends only on a initial and final points and not on the path chosen.

Q. 16 Prove that line integral of a conservative field over a closed path is zero.

Answer: From above question,

$\oint{\vec{A}.\vec{dr}} = - \int_{A}^{B}d\phi = - (\phi_{A} -\phi_{B}) =\phi_{B}-\phi_{A}$

For closed path A=B, so

$\oint{\vec{A}.\vec{dr}} = A - \phi_{A} = 0$ , as required.

Q.17 Show thatfor an incompressible fluid div *= 0.***

Answer: By definition of divergence,

$Div\vec{A} = \frac{\partial A_x}{\partial x}\ +\ \frac{\partial A_y}{\partial y}+\frac{\partial A_z}{\partial z}$

In certain region where quantity of fluid flowing outward is equal to flowing inward, we can write

$\left[\frac{\partial A_x}{\partial x}\ +\ \frac{\partial A_y}{\partial y}+\frac{\partial A_z}{\partial z}\right]dxdydz = - \frac{\partial}{\partial t\ }\rho dxdydz$

where is density of fluid.

$\left[\frac{\partial A_x}{\partial x}\ +\ \frac{\partial A_y}{\partial y}+\frac{\partial A_z}{\partial z}\right] \left(dxdydz\right) = - \frac{\partial\rho}{\partial t\ }dxdydz$

$\left[\frac{\partial A_x}{\partial x}\ +\ \frac{\partial A_y}{\partial y}+\frac{\partial A_z}{\partial z}+\ \frac{\partial\rho}{\partial t\ }\right] dxdydz = 0$

$\left[Div\vec{A}+\ \frac{\partial\rho}{\partial t\ }\right] dxdydz = 0$

$Div\vec{A}+\ \frac{\partial\rho}{\partial t\ } = 0$ , If fluid is incompressible

Q.18 Define Laplacian operator and hence Laplace equation.

Answer: = $\nabla^{2\ }= \vec{\nabla.} \vec{\nabla} \rightarrow \nabla^{2\ }=\left[\ \hat{i}\frac{\partial}{\partial x}+\ \hat{j}\frac{\partial}{\partial y}+\hat{k}\frac{\partial}{\partial z}\right] . \left[\ \hat{i}\frac{\partial}{\partial x}+\ \hat{j}\frac{\partial}{\partial y}+\hat{k}\frac{\partial}{\partial z}\right]$

${\ \nabla}^{2\ } = {\frac{\partial}{\partial x}}^2+\ \frac{\partial^2}{\partial y}+{\frac{\partial}{\partial z}}^2$

Laplace equation is,

$\Delta ^{2}\phi =0$

Q.19 What is meant by irrotational and solenoidal fields?

Answer; Irrotational field; A vector whose curl is zero is called irrotational field or conservative field.

Electric field is an example of irrotational field.

Solenoidal field; A vector field whose divergence vanishes is known as solenoidal field.

Magnetic field is an example of solenoidal field since divB =0 .

Q.20 State various conditions for a vector field to be conservative.

Answer: Different conditions for a vector field to be conservative are;

The line integral of a conservative field over a closed path is zero i.e.

$\oint\vec{A}.\vec{dr} = 0$

The curl of the conservative field is zero. curl $\vec{A}$ =0
A conservative field $\vec{A}$ is derivable from a scalar field $\phi$ .

Q.21 Give two examples of vector point function and their divergence what does describe?

Answer: Examples of vector point functions are electric field vector $\vec{E}$ and velocity vector $\vec{V}$ . The divergence of such vector point functions describes respective vector fields.

Q.22 Time is always directed from present to future. Is it a vector quantity?

Answer: Time is not vector quantity because present to future does not represent direction any specified direction as compared to direction indicator or reference axis.

Q.23 If sum of two vectors is perpendicular to their difference, what will be the relation in their magnitudes?

Answer: It is given that,

$(\vec{A} + \vec{B} ) (\vec{A} - \vec{B} ) = 0 \rightarrow \vec{A}.\vec{A} +\vec{A}.\vec{B}- \vec{B}.\ \vec{A} -\vec{B}.\ \vec{B} = 0$

i.e. $\vec{A}$ and $\vec{B}$ have same magnitudes.

Q.24 Given that A + B = C & A² + B² = C², how A and B are oriented relative to each other?

Answer: From the given conditions, it is clear that $\vec{A}$ + $\vec{B}$ and $\vec{C}$ form a right angle triangle with $\vec{C}$ as hypotenuse, hence A and B are orthogonal.

Q.25 Show that the resultant of two vectors of same magnitude bisect the angle between them?

Answer: Using,

$tan\phi = \frac{B\sin{\theta}}{A+B\cos{\theta}} = \frac{A\sin{\theta}}{A+A\cos{\theta}} = \frac{\sin{\theta}}{1+\cos{\theta}}$

$tan\phi = \frac{2sin\ \frac{\theta}{2}\ cos\ \frac{\theta}{2}}{2{cos}^{2\ }\frac{\theta}{2}} = tan \frac{\theta}{2}$

$\phi=\frac{\theta}{2}$ , as required.

Q.26 Consider a vector with non-zero components A_xand A_y such that A_x+ A_y =0 . Find possible directions of .

Answer: It is given that A_x= – A_y≠ 0 , so $\theta = tan^{-1}\left[\frac{A_y}{A_x}\right] = tan^{-1} \left[-\frac{A_y}{A_x}\right] = 135^o$

Hence $\vec{A}$ vector is oriented at 135° from x- axis in anticlockwise direction or 45° in clockwise direction.

Q.27 Explain how two vectors of same magnitude would have to be oriented if they were to be combined to have magnitude of full resultant vector?

Answer: Using, $R = \sqrt{A^2+B^2+2AB\ cos\theta}$

Put A= B and R=0 to get,

$0 = \sqrt{A^2+A^2+2AA\ cos\theta} \ \ \ \rightarrow 2A^2+2A^2 + cos\ \theta = 0$

$2A2cos\theta = -2A2 \rightarrow cos \theta = -1 \rightarrow \theta = 180^o$

Q.28 If x =0 , can it be concluded that and are parallel to each other?

Answer: If $\vec{A}$ x $\vec{B}$ =0 then it can be concluded that $\vec{A}$ and $\vec{B}$ are parallel or anti parallel to each other provided $\vec{A}$ and $\vec{B}$ are not null vectors.

Q.29 Calculate vector product of vectors = 2 and . What will be direction of x .

Answer: He should hold his umbrella in the forward direction so that it points opposite to relative velocity of rain.

Q.30 Torque is defined as x . If the line of the action of the applied force passes through the pivot point, then what is the value of torque?

Answer: When line of action of force passes through the pivot, the moment arm will be zero, so torque is zero.

Q.31 Rain is falling vertically downward and a student is running for shelter. To keep driest in which direction he should hold his umbrella?

Answer: He should hold his umbrella in the forward direction so that it points opposite to relative velocity of rain.

Q.32 Write a formula to find unit vectors perpendicular to and lying in plane of

Answer: Unit vectors perpendicular to A and lying in plane of B and C are,

\hat{n} = \pm \frac{\vec{A\ }\ \ x\ (\vec{B}x\ \vec{C})}{\left|\vec{A\ }\ \ x\ (\vec{B}x\ \vec{C})\right|}

Q.33 Differentiate between Curl and rotation?

Answer: The rotation of maximum value is called curl. When rotation is clockwise, curl is directed into plane of paper according to right hand rule.

Q.34 What is the relation between line integral and surface integral?

Answer: The relation between line integral and surface integral is given by an important theorem, called Stoke’s theorem which states that;

“The line integral of a tangential component of a vector A taken around a simple closed curve C is equal to surface integral of normal component of curl of vector A taken over any surface S having C as its boundary.

Mathematically,

$\oint\vec{A}.\vec{dl} = \int_{S}^{1}{curl\ \vec{A}}.d\vec{s}$

Q.35 Give physical definition of gradient. Also mention direction of gradient of a scalar field?

Answer: Gradient S is in direction in which the change in S is most rapid and its magni

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Mechanics-I

Q.1 Can a scalar product of two vectors be negative? If your answer is yes, give an example. If not, give proof.

Q.2 A plane is moving at 100km/h at an angle of 60° with ground. What is speed of its projection on ground if the sun is just above plane at noon?

Q.3 Consider a vector A with components A_xand A_y < 0.Find possible directions of vector A .

Q.4 Under what conditions would a vector have components equal in magnitude?

Q.5 Does the value of vector quantity depends upon selected coordinate system?

Q.6 Weather gradient of a scalar field depends upon any particular system of coordinates?

Q.7 Suppose C= A+ B, does it follow that either C ≥ A or C ≥ B? If not, why? If yes prove it.

Q.8 Suppose is non-zero vector. It is given that AXB=0 and . A.B=0. What can you conclude about B.

Q.9 If all components of vectors and are reversed, how will this affect

O.10 A vector lies in xy-plane. For what orientation will both rectangular components be negative? For what orientation will its components have opposite signs?

Q.11 Scalar triple product is cyclic. Can you give such a statement for vector triple product?

Q. 12 Can we say that cylindrical polar coordinates are orthogonal?

Q.13 What is the physical significance of derivative?

Q.14 What is geometrical interpretation of gradient of a scalar field?

Q.15 You know that “ line integral of a conservative field is independent of path”. Prove this statement.

Q. 16 Prove that line integral of a conservative field over a closed path is zero.

Q.17 Show thatfor an incompressible fluid div *= 0.***

Q.18 Define Laplacian operator and hence Laplace equation.

Q.19 What is meant by irrotational and solenoidal fields?

Q.20 State various conditions for a vector field to be conservative.

Q.21 Give two examples of vector point function and their divergence what does describe?

Q.22 Time is always directed from present to future. Is it a vector quantity?

Q.23 If sum of two vectors is perpendicular to their difference, what will be the relation in their magnitudes?

Q.24 Given that A + B = C & A² + B² = C², how A and B are oriented relative to each other?

Q.25 Show that the resultant of two vectors of same magnitude bisect the angle between them?

Q.26 Consider a vector with non-zero components A_xand A_y such that A_x+ A_y =0 . Find possible directions of .

Q.27 Explain how two vectors of same magnitude would have to be oriented if they were to be combined to have magnitude of full resultant vector?

Q.28 If x =0 , can it be concluded that and are parallel to each other?

Q.29 Calculate vector product of vectors = 2 and . What will be direction of x .

Q.30 Torque is defined as x . If the line of the action of the applied force passes through the pivot point, then what is the value of torque?

Q.31 Rain is falling vertically downward and a student is running for shelter. To keep driest in which direction he should hold his umbrella?

Q.32 Write a formula to find unit vectors perpendicular to and lying in plane of

Q.33 Differentiate between Curl and rotation?

Q.34 What is the relation between line integral and surface integral?

Q.35 Give physical definition of gradient. Also mention direction of gradient of a scalar field?

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Mechanics-I

Q.1 Can a scalar product of two vectors be negative? If your answer is yes, give an example. If not, give proof.

Q.2 A plane is moving at 100km/h at an angle of 60° with ground. What is speed of its projection on ground if the sun is just above plane at noon?

Q.3 Consider a vector A with components Ax and Ay < 0.Find possible directions of vector A .

Q.4 Under what conditions would a vector have components equal in magnitude?

Q.5 Does the value of vector quantity depends upon selected coordinate system?

Q.6 Weather gradient of a scalar field depends upon any particular system of coordinates?

Q.7 Suppose C= A+ B, does it follow that either C ≥ A or C ≥ B? If not, why? If yes prove it.

Q.8 Suppose is non-zero vector. It is given that AXB=0 and . A.B=0. What can you conclude about B.

Q.9 If all components of vectors and are reversed, how will this affect

O.10 A vector lies in xy-plane. For what orientation will both rectangular components be negative? For what orientation will its components have opposite signs?

Q.11 Scalar triple product is cyclic. Can you give such a statement for vector triple product?

Q. 12 Can we say that cylindrical polar coordinates are orthogonal?

Q.13 What is the physical significance of derivative?

Q.14 What is geometrical interpretation of gradient of a scalar field?

Q.15 You know that “ line integral of a conservative field is independent of path”. Prove this statement.

Q. 16 Prove that line integral of a conservative field over a closed path is zero.

Q.17 Show that for an incompressible fluid div = 0.

Q.18 Define Laplacian operator and hence Laplace equation.

Q.19 What is meant by irrotational and solenoidal fields?

Q.20 State various conditions for a vector field to be conservative.

Q.21 Give two examples of vector point function and their divergence what does describe?

Q.22 Time is always directed from present to future. Is it a vector quantity?

Q.23 If sum of two vectors is perpendicular to their difference, what will be the relation in their magnitudes?

Q.24 Given that A + B = C & A2 + B2 = C2, how A and B are oriented relative to each other?

Q.25 Show that the resultant of two vectors of same magnitude bisect the angle between them?

Q.26 Consider a vector with non-zero components Ax and Ay such that Ax + Ay =0 . Find possible directions of .

Q.27 Explain how two vectors of same magnitude would have to be oriented if they were to be combined to have magnitude of full resultant vector?

Q.28 If x =0 , can it be concluded that and are parallel to each other?

Q.29 Calculate vector product of vectors = 2 and . What will be direction of x .

Q.30 Torque is defined as x . If the line of the action of the applied force passes through the pivot point, then what is the value of torque?

Q.31 Rain is falling vertically downward and a student is running for shelter. To keep driest in which direction he should hold his umbrella?

Q.32 Write a formula to find unit vectors perpendicular to and lying in plane of

Q.33 Differentiate between Curl and rotation?

Q.34 What is the relation between line integral and surface integral?

Q.35 Give physical definition of gradient. Also mention direction of gradient of a scalar field?

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Q.3 Consider a vector A with components A_xand A_y < 0.Find possible directions of vector A .

Q.17 Show thatfor an incompressible fluid div *= 0.***

Q.24 Given that A + B = C & A² + B² = C², how A and B are oriented relative to each other?

Q.26 Consider a vector with non-zero components A_xand A_y such that A_x+ A_y =0 . Find possible directions of .