Deck 2: Vector Analysis

Question 1

Three vectors A, B, and C, drawn in a head-to-tail fashion, form three sides of a triangle. What is A + B + C What is A + B C

Answer 1

Draw vector diagram using three vectors
in a head to tail fashion:
Figure 1
In Head-to-tail rule, the value of sum of two vectors
is
. But when we draw the three vectors
in head-to-tail rule forms the three sides of a triangle is shown in Figure 1. When
is in opposite direction. The value of sum of two vectors
is
.
Calculate the value of sum of vectors,
.
Therefore, the value of vector
is
.
Calculate the value of sum of vectors,
.
Therefore, the value of vector
is
.

Question 2

Given two points P 1 ( 2, 0, 3) and P 2 (0, 4, 1), find
a) the length of the line joining P 1 and P 2 , and
b) the perpendicular distance from the point P 3 (3, 1, 3) to the line.

Answer 2

(a)
Consider the points
as
.
Write the expression for the vector,
.
Write the expression for the vector,
.
Calculate the vector,
.
Calculate the magnitude of the vector,
.
Therefore, the length of the line joining
,
is
.
(b)
Consider a point
as shown in Figure 1 and draw a perpendicular from this point to line
which meets at point
.
Figure 1
The perpendicular distance from the point
to the line is
, which equals
.
Therefore,
…… (1)
Write the expression for the vector,
.
Calculate the vector,
.
Substitute the known values in equation (1).
Therefore, the perpendicular distance from
to the line,
is
.

Question 3

What are the expressions for A · B and A × B in Cartesian coordinates

Answer 3

Write expressions for vectors
and
in Cartesian coordinates.
Calculate the expression for
in Cartesian coordinates.
Therefore, the expression for
in Cartesian coordinates is
.
Calculate the expression for
in Cartesian coordinates.
Therefore, the expression for
in Cartesian coordinates is
.

Question 4

State the divergence theorem in words.

Question 5

A rhombus is an equilateral parallelogram. Denote two neighboring sides of a rhombus by vectors A and B.
a) Verify that the two diagonals are A + B and A B.
b) Prove that the diagonals are perpendicular to each other.

Question 6

Does A × B = A × C imply B = C Explain.

Question 7

Given a vector field A = a r r + a z z ,
a) find the total outward flux over a circular cylinder around the z-axis with a radius 2 and a height 4 centered at origin.
b) Repeat (a) for the same cylinder with its base coinciding with the xy -plane.
c) Find · A and verify the divergence theorem.

Question 8

What is the physical definition of the curl of a vector field

Question 9

Under what conditions can the dot product of two vectors be negative

Question 10

The cylindrical coordinates of two points P 1 and P 2 are: P 1 (4, 60°, 1) and P 2 (3, 180°, 1). Determine the distance between these two points.

Question 11

Find the results of the following products of unit vectors:
a) a , · a x ,
b) a R · a y ,
c) a z · a R ,
d) a × a x ,
e) a r × a R ,
f) a × a z.

Question 12

State Stokes's theorem in words.

Question 13

Compare the values of the following scalar triple products of vectors:
(a) (A × C)·B ,
(b) A·(C × B) ,
(c) (A × B)·C , and
(d) B·(a A × A).

Question 14

Given vector A = a x 5 a y 2 + a z , find the expression of
a) a unit vector a B such that a B || A, and
b) a unit vector a C in the xy -plane such that a C A.

Question 15

What is the difference between a scalar quantity and a scalar field Between a vector quantity and a vector field

Question 16

Find the divergence of the following radial vector fields:
a) f 1 ( R ) = a R R n ,
b)

Question 17

If the three sides of an arbitrary triangle are denoted by vectors A, B, and C in a clockwise or counterclockwise direction, then the equation A + B + C = 0 holds. Prove the law of sines.
HINT: Cross multiply the equation separately by A and by B , and examine the magnitude relations of the products.

Question 18

What makes a coordinate system (a) orthogonal and (b) right-handed

Question 19

Find the clockwise circulation of the vector field F given in Example 2-14 around a square path in the xy -plane centered at the origin and having four units on each side ( 2 x 2 and 2 y 2).
Example

Question 20

What is the difference between an irrotational field and a solenoidal field

Question 21

Write down the results of A · B and A × B if (a) A || B , and (b) A B.

Question 22

Transform Cartesian coordinates (4, 6, 12) into spherical coordinates.

Question 23

Express the r-component, A r , of a vector A at ( r 1 1 , z 1 )
a) in terms of A x and A y in Cartesian coordinates, and
b) in terms of A R and A in spherical coordinates.

Question 24

State Helmholtz's theorem in words.

Question 25

Which of the following expressions do not make sense
(a) A × B/|B| ,
(b) C · D/(A × B) ,
(c) AB/CD ,
(d) A × B/(C · D) ,
(e) ABC,
(f) A × B × C.

Question 26

Decompose vector A = a x 2 a y 5+ a z 3 into two components, A 1 and A 2 , that are, respectively, perpendicular and parallel to another vector B = a x + a y 4.

Question 27

What is the physical definition of the gradient of a scalar field

Question 28

For vector function A = a r r 2 + a z 2 z , verify the divergence theorem for the circular cylindrical region enclosed by r = 5, z = 0, and z = 4.

Question 29

Given three vectors A, B, and C as follows:
A = a x 6 + a y 2 a z 3,
B = a x 4 a y 6 + a z 12,
C = a x 5 a z 2,
find
a) a B ,
b) |B A|,
c) the component of A in the direction of B ,
d) B · A,
e) the component of B in the direction of A ,
f) AB ,
g) A × C, and
h) A · (B × C) and (A × B) · C.

Question 30

What are metric coefficients

Question 31

Given F = a r sin + a 3 cos and the quarter-circular region shown in Fig. 2-22,
a) determine
F · d , and
b) find × F , and verify Stokes's theorem.
Fig Path for line integral (Example 2-14 and 2-16)
Example 1
Example 2

Question 32

A vector field D = a R (cos 2 )/ R 3 exists in the region between two spherical shells defined by R = 1 and R = 2. Evaluate
a)

b) • D d v.

Question 33

Is (A · B)C equal to A(B · C) Explain.

Question 34

Derive the formula for the surface of a sphere with a radius R 0 by integrating the differential surface area in spherical coordinates.

Question 35

Express the -component, E , of a vector E at ( R 1 , l , l )
a) in terms of E x , E y , and E z in Cartesian coordinates, and
b) In terms of E r and E z in cylindrical coordinates.

Question 36

For a scalar function f and a vector function A, prove that
• ( f A ) = f • A + A • f
in Cartesian coordinates.

Question 37

Given a vector B = a x 2 a y 6 + a z 3, find
a) the magnitude of B ,
b) the expression for a B ,
c) the angles that B makes with the x, y , and z axes.

Question 38

Equation (2-15) in Example 2-2 describes the scalar triple products of three vectors A, B, and C. There is another important type of product of three vectors. It is a vector triple product, A × (B × C). Prove the following relation by expansion in Cartesian coordinates:
A × (B × C) = B(A · C)-C(A · B). (2-113)
Equation (2-113) is known as the "BAC-CAB" rule.

Question 39

Express the space rate of change of a scalar in a given direction in terms of its gradient.

Question 40

Let unit vectors a A and a B denote the directions of vectors A and B in the xy -plane that make angles and , respectively, with the x -axis.
a) Obtain a formula for the expansion of the cosine of the difference of two angles, cos ( ), by taking the scalar product a A · a B.
b) Obtain a formula for sin ( ) by taking the vector product a B × a A.

Question 41

Write d and dv (a) in Cartesian coordinates, (b) in cylindrical coordinates, and (c) in spherical coordinates.

Question 42

Prove the identity in Eq.(2-105) in Cartesian coordinates.

Question 43

Given two vectors A and B , how do you find (a) the component of A in the direction of B , and (b) the component of B in the direction of A

Question 44

Assuming V = xy 2 yz, find, at point P (2, 3, 6),
a) the direction and the magnitude of the maximum increase of V, and
b) the space rate of decrease of V in the direction toward the origin.

Question 45

Given a vector field in spherical coordinates F = a R (12/ R 2 ).

Question 46

Given two points P 1 (l, 2, 0) and P 2 ( 3, 4, 0) in Cartesian coordinates with origin O , find
a) the length of the projection of
on
and
b) the area of the triangle OP 1 P 2.

Question 47

Find the component of the vector A = a x z a z x at the point P 1 ,( 1, 0, 2) that is directed toward the point
.

Question 48

What is the expression for the del operator, , in Cartesian coordinates

Question 49

The three corners of a right triangle are at P 1 (1, 0, 2), P 2 ( 3, 1, 5), and P 3 (3, 4, 6).
a) Determine which corner is a right angle.
b) Find the area of the triangle.

Question 50

10 Given two points P 1 (l, 2, 3) and P 2 ( 1, 0, 2) in Cartesian coordinates, write the expressions of the vectors
and

Question 51

Prove the identity in Eq. (2-109) in Cartesian coordinates.

Question 52

Does A · B = A · C imply B = C Explain.

Question 53

Solve Example 2-10 in cylindrical coordinates.
Example 2-10

Question 54

What is the physical definition of the divergence of a vector field

Question 55

Express the position vector
from the origin O to the point Q (3, 4, 5) in cylindrical coordinates.

Question 56

The position of a point in cylindrical coordinates is given by (3, 4 /3, 4). Specify the location of the point
a) in Cartesian coordinates, and
b) in spherical coordinates.

Question 57

Determine wether the following vector fields are irrotational, solenoidal, both, or neither:
a) A = a x xy a y y 2 + a z xz,
b) B = r ( a r sin + a 2 cos ) ,
c) C = a x x a y 2 y + a z z,
d) D = a R k/R.