University of Missan
College of Engineering
Electrical Engineering
Department
1st Semester Year
2013-2014
2nd Lesson Stage
Engineering Electromagnetic Fields
Subject: Vector Analysis
Tutorial Sheet No. 1
Dr. Ahmed Thamer Radhi
2013 - 2014
Electromagnetic Fields
Vector Analysis
Tutorial Sheet No.1
Lesson Year 1 st Semester:2013-2014
Stage
2 nd Year
Subject
Tutorial
Tutorial No.
1
Lecturer
Dr. Ahmed Thamer
University of Missan
College of Engineering
Electrical Engineering Dept.
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Engineering
Electromagnetics Fields
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�⃗= 5 ā x + 2 ā y - 6 ā z ; determine:
Q1] Given Vectors �A⃗= ā x + 3ā z and B
�⃗ + B
�⃗ - B
�⃗ along a unit vector parallel to (3A
�⃗ + B
�⃗|
�⃗)
�⃗)
1- |A
2- (5A
3- Component of A
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Q2] Given points P (1, -3, 5), Q (2, 4, 6) & R (0, 3, 8). Find the position vectors of P and R,
the distance vector r⃗ QP and the distance between Q & R.
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�⃗ =2ā x + 5ā y + 4ā z , B
�⃗= ā x - 6ā z ; Calculate:
�⃗= 3ā x + ā y + 5ā z and C
Q3] For Vectors A
�⃗ 2- �A⃗xB
�⃗ 3- �A⃗.(B
�⃗ - �C⃗) 4- �A⃗x(B
�⃗ + �C⃗)
1- �A⃗.B
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Q4] Given the two Vectors r⃗ A = -ā x - 3ā y - 4ā z , r⃗ B = 2ā x + 2ā y + 2ā z and point C(1, 3, 4), find:
1- �R⃗ AB
2- |r⃗ A |
3- a�⃗ A
4- a�⃗ AB
5- a unit vector directed from C toward A.
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�⃗= 5ā x + 2ā y - 6ā z ; Find θ AB .
Q5] If �A⃗= ā x + 3ā z and B
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�⃗ and
�⃗= 4α ā x + 8 ā y - 2α ā z , for what value or values of α are A
Q6] If �A⃗= α ā x + 2ā y + 10ā z and B
�⃗ perpendicular?
B
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Q7] Let �E⃗= 3 ā y + 4 ā z , �F⃗= 4 ā x – 10 ā y + 5 ā z . Find: (a) the scalar and vector components of �E⃗
along �F⃗ (b) a unit vector perpendicular to both �E⃗ and �F⃗.
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�⃗= 2ā x + 3ā y - 4ā z and B
�⃗xB
�⃗= -6ā x - 4ā y + ā z ; find scalar and vector components of A
�⃗
Q8] If A
�⃗= ā x - ā y + ā z .
along the direction of vector C
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�⃗ and B
�⃗ =(B x, B y, 3); find B x and B y such that A
�⃗ are parallel.
Q9] If �A⃗ =(1, 2, -1) and B
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Q10] Given �A⃗= 3ā x - 4ā y + 7ā z , find:
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abcde-
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�⃗
āx X A
�A⃗ . (ā y X ā x )
(ā x X ā z ) X �A⃗
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(ā x X ā y )X(ā z X �A⃗)
�⃗ X ā y ) . ā z
(A
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Q11] Given points P (ρ =6, Ф=125 o , z=-3), and Q (x=3, y=-1, z=4), find the distance from:
a- P to the origin
b- Q perpendicular to z-axis
c- P to Q.
Dr. Ahmed Thamer
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Vector Analysis
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Electromagnetic Fields
Vector Analysis
���⃗=(x-y) ā y in cylindrical coordinates system.
Q12] (a) Express the vector field W
(b) Express the vector field �F⃗= ρ cosФ ā ρ in Cartesian coordinates system.
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Tutorial Sheet No.1
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Q13] Given points P (r=6, θ =110 o , Ф=125 o ), and Q (x=3, y=-1, z=4), find the distance from:
a- Q to the origin
b- P perpendicular to y-axis
c- P to Q.
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���⃗=(x-y) ā y in spherical coordinates system.
Q14] (a) Express the vector field W
�⃗= r cosФ ā r in Cartesian coordinates system.
(b) Give the vector field F
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Q15] Given the vector field �F⃗= 0.4(y-2x) ā x – [200/(x 2 +y 2 +z 2 )] ā z :
�⃗| at P (- 4, 3, 5).
1- Evaluate |F
�⃗ at P.
2- Find a unit vector specifying the direction of F
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Q16] Given points A(2, 5, -1), B(3, -2, 11) and C(-2, 3, 1), find:
�⃗ AB .R
�⃗ AC ).
�⃗ AB and R
�⃗ AC . 3- The length of the
1- (R
2- The angle between R
�⃗ AB on �R⃗ AC .
4- The vector projection of �R⃗ AB on �R⃗ AC .
projection of R
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Q17] A triangle is defined by the three points A(2, -5, 1), B(-3, 2, 4) and C(0, 3, 1), find:
�⃗ BC X �R⃗ BA ).
1- (R
2- The area of the triangle.
3- A unit vector perpendicular
to the plane in which the triangle is located.
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Q18] Obtain a vector of magnitude is (10) that has the same direction as the vector orthogonal
�⃗= 2ā x - 4ā y + 3ā z and B
�⃗= 2ā x + 5ā y ?
to both A
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�⃗= u x ā x + 5ā y - ā z , V
�⃗= 2ā x – v y ā y + 3ā z , and W
���⃗= 6ā x + ā y + w z ā z , obtain u x, v y,
Q19] For U
and
�⃗, �V⃗, and W
���⃗ are mutually orthogonal.
w z such that U
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Q20] Verify that:
12-
�A⃗ . (A
�⃗ X B
�⃗ X B
�⃗) = B
�⃗ . (A
�⃗) = 0
�⃗.B
�⃗ X �B⃗| 2 = (|A
�⃗| |B
�⃗) 2 + |A
�⃗|) 2
(A
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�⃗= 4ā x - 2ā y - ā z and B
�⃗= ā x - 4ā y - 4ā z are orthogonal?
Q21] Prove that: A
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�⃗= 2ā x + 8ā y + 3ā z are represented by directed line
Q22] The vectors �A⃗= 4ā x + 5ā y - 2ā z and B
segments that extended outward from the origin of a Cartesian coordinate system.
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1- What is the separation of their tips?
�⃗?
2- Find a unit vector in the direction of A
�⃗ that is parallel to A
�⃗ and has the length of B
�⃗?
3- Find a vector C
Dr. Ahmed Thamer
Vector Analysis
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Electromagnetic Fields
Vector Analysis
Tutorial Sheet No.1
Q23] (a) Determine the angle between (Ax ā x - 7ā y + 4ā z ) and (5āx + 4ā y - 3ā z ) , if A x =10.
(b) What is the value of A x if the angle is to be: (I- 90 o II- 62.1 o )?
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�⃗= yā x + (x+z) ā y. Express P and A
�⃗ in cylindrical
Q24] Given point P(-2, 6, 3) and vector A
coordinates. Evaluate �A⃗ at P in Cartesian, cylindrical and spherical systems.
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Q25] Given points A(x=2, y=3, z=-1) and B(ρ =4, Ф= -50 o , z=2), find the distance from:
a- A to the origin
b- B to the origin
c- A to B
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Q26] Express the following vectors in Cartesian coordinates:
a- �F⃗= ρz sinФ ā ρ + 3ρ cosФ ā Ф + ρ cosФ sinФ āz
�⃗= ρ 2 ā ρ + sinθ ā Ф
b- E
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Q27] Transform each of the following vectors to cylindrical coordinates at the points specified:
a) 5ā x at P(ρ =4, Ф= 120 o , z=2)
b) 5ā x at R(x=3, y=4, z=-1)
c) 4ā x - 2ā y - 4ā z at Q(x=2, y=3, z=5)
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Q28] Given the points A(x=2, y=3, z= -1) and B(r=4, θ =25 o , Ф=120 o ), find:
a) The spherical coordinates of A.
b) The Cartesian coordinates of B.
c) The distance from A to B.
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�⃗= ρz cosФ ā ρ + e -z sinФ ā Ф + ρ 2 ā z at point (1, π , 0), find:
Q29] Given the vector field �H
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�⃗ . ā x
(a) �H
��⃗ normal to surface ρ=1
(b) The vector component of H
�⃗ tangential to the plane z=0
(c) The scalar component of �H
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Q30] Transform each of the following vectors to spherical coordinates at the points specified:
a) 5ā x at A(r=4, θ =25 o, Ф=120 o )
b) 5ā x at B(x=2, y=3, z=-1)
c) 4ā x - 2ā y - 4ā z at C(x= -2, y= -3, z=4)
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Q31] Using the appropriate differential elements, show that:
a- The circumference of a circle of radius (r o ) is 2πr o ?
b- The surface area of a sphere of radius (r o ) is 4πr o 2 ?
c- The volume of a sphere of radius (r o ) is 4/3πr o 3 ?
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Q32] A three-dimensional solid is described in spherical coordinates according to:
π
0≤ r ≤1, 0≤ θ ≤ , 0≤ Ф ≤2π
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a- Sketch the solid.
b- Determine the volume of the solid.
c- Determine the surface of the solid.
Dr. Ahmed Thamer
Vector Analysis
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Electromagnetic Fields
Vector Analysis
Tutorial Sheet No.1
Q33] Given a vector field expressed in mixed coordinate variables as:
2
⃗J= 3xz2 ā x + 3ycos 𝜃𝜃 ā y +(2- 3y22 - 3x2 ) ā z
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Express ⃗J completely in spherical coordinate?
Q34] Using cylindrical coordinates to find the area of a shell of a cylinder where z=5, ρ=2, and
30 o ≤ Ф ≤120 o?
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���⃗= -10ā x + 4ā y - 8ā z and N
��⃗= 8ā x + 7ā y - 2ā z , find:
Q35] Given the vectors M
���⃗ + 2N
��⃗
a- A unit vector in the direction of -M
��⃗ - 3M
���⃗
b- The magnitude of 5ā x + N
���⃗||2N
��⃗|(M
���⃗ + N
��⃗)
c- |M
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Q36] The vector from the origin to point A is given as 6ā x - 2ā y - 4ā z , and the unit vector
directed from the origin toward point B is (2/3, -2/3, 1/3). If points A and B are 10 units
apart, find the coordinates of point B?
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Q37] Express the unit vector ā x in spherical components at the points:
a- r=2, θ=1 rad, Ф=0.8 rad
b- x=3, y=2, z= -1
c- ρ=2.5, Ф=0.7 rad, z=1.5
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Q38] Given A(r=20, θ=30 o , Ф=45 o ) and B(r=30, θ=115 o , Ф=160 o), find:
�⃗ AB |
a- |R
�⃗ AC |, given C(r=20, θ=90 o , Ф=45 o )
b- |R
c- The distance from A to C
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�⃗= y ā x – 2.5x ā y + 3ā z , and the point Q(4, 5, 2), find:
Q39] Given the vector field G
(a) �G⃗ at Q.
(b) Scalar component of �G⃗ at Q in the direction of unit vector a�⃗ N .
�⃗ at Q in the direction of unit vector a�⃗ N .
(c) Vector component of G
(d) The angle between �G⃗ at Q and �a⃗ N .
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Q40] The three vertices of a triangle are located at A(6, -1, 2), B(-2, 3, -4), and C(-3, 1, 5), find:
(a) �R⃗ AB
(b) �R⃗ AC
(c) The angle θ BAC at vertex A
�⃗ AB on R
�⃗ AC
(d) The vector projection of R
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&&&& Best Wishes &&&&
Dr. Ahmed Thamer
Vector Analysis
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Electromagnetic Fields
Vector Analysis
Dr. Ahmed Thamer
Vector Analysis
Tutorial Sheet No.1
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