Elements Of Electromagnetics
Elements Of Electromagnetics
7th Edition
ISBN: 9780190698614
Author: Sadiku, Matthew N. O.
Publisher: Oxford University Press
bartleby

Videos

Question
Book Icon
Chapter MA, Problem 1.1MA

(i)

To determine

To obtain: The value of θ from the inequality |A+B|<|AB|.

(i)

Expert Solution
Check Mark

Answer to Problem 1.1MA

The angle between the two vectors A and B  is θ>90°.

Explanation of Solution

The angle between the two vectors A and B is θ.

Suppose the inequality |A+B|<|AB| exists.

This implies the magnitude of AB is greater than the magnitude of A+B.

It further implies that the vector B of higher magnitude when subtracted from the vector A of lower magnitude, the resultant vector with greater magnitude is obtained.

However, it’s not just the magnitude but the signs of the components of the vectors.

When the vector B of opposite sign to that of the vector A, is subtracted from A; it adds to the value of A, hence magnitude of the resultant vector increases

When the vector B of opposite sign to that of the vector A, is added to A; it reduces the value of A, hence the magnitude of the resultant vector decreases.

Due to change in signs of the components of two vectors, the angle between the two vectors is always greater than 90°.

Example: A=3ax+2ay and B=ax3ay

Both the vectors lie in different quadrant, hence the angle between them is greater than 90°.

A+B=3ax+2ayax3ay=2axay|A+B|=22+(1)2=4+1=5

AB=3ax+2ay(ax3ay)=4ax+5ay|AB|=42+52=16+25=41

Thus, the inequality |A+B|<|AB| is verified.

Therefore, when |A+B|<|AB|, the angle between the two vectors A and B  is θ>90°.

(ii)

To determine

To obtain: The value of θ from the inequality |A+B|=|AB|.

(ii)

Expert Solution
Check Mark

Answer to Problem 1.1MA

The angle between the two vectors A and B  is θ=90°.

Explanation of Solution

The angle between the two vectors A and B is θ.

Suppose the inequality |A+B|=|AB| exists.

Example: A=ax and B=ay

Both the vectors are on the axis, hence the angle between them is 90°.

A+B=ax+ay|A+B|=12+12=1+1=2

AB=axay|AB|=12+(1)2=1+1=2

Thus, the inequality |A+B|=|AB| is verified.

Therefore, when |A+B|=|AB|, the angle between the two vectors A and B  is 90°.

(i)

To determine

To obtain: The value of θ from the inequality |A+B|<|AB|.

(i)

Expert Solution
Check Mark

Answer to Problem 1.1MA

The angle between the two vectors A and B  is θ<90°.

Explanation of Solution

The angle between the two vectors A and B is θ.

Suppose the inequality |A+B|>|AB| exists.

This implies the magnitude of A+B is greater than the magnitude of AB.

It further implies that the vector B when subtracted from the vector A, the resultant vector with greater magnitude is obtained.

However, it’s not just the magnitude but the signs of the components of the vectors.

When the vector B of same sign to that of the vector A, is added to A; it adds to the value of A, hence magnitude of the resultant vector increases

When the vector B of same sign to that of the vector A, is subtracted from A; it reduces the value of A, hence the magnitude of the resultant vector decreases.

As the sign of the components of two vectors are same, the angle between the two vectors is always less than 90°.

Example: A=3ax+2ay and B=ax+3ay

Both the vectors lie in same quadrant, hence the angle between them is less than 90°.

A+B=3ax+2ayax+3ay=4ax+5ay|A+B|=(4)2+52=16+25=41

AB=3ax+2ay+ax3ay=2axay|AB|=(2)2+(1)2=4+1=5

Thus, the inequality |A+B|>|AB| is verified.

Therefore, when |A+B|>|AB|, the angle between the two vectors A and B  is θ<90°.

Want to see more full solutions like this?

Subscribe now to access step-by-step solutions to millions of textbook problems written by subject matter experts!
Students have asked these similar questions
Q Derive (continuity equation)? I want to derive clear mathematics.
motor supplies 200 kW at 6 Hz to flange A of the shaft shown in Figure. Gear B transfers 125 W of power to operating machinery in the factory, and the remaining power in the shaft is mansferred by gear D. Shafts (1) and (2) are solid aluminum (G = 28 GPa) shafts that have the same diameter and an allowable shear stress of t= 40 MPa. Shaft (3) is a solid steel (G = 80 GPa) shaft with an allowable shear stress of t = 55 MPa. Determine: a) the minimum permissible diameter for aluminum shafts (1) and (2) b) the minimum permissible diameter for steel shaft (3). c) the rotation angle of gear D with respect to flange A if the shafts have the minimum permissible diameters as determined in (a) and (b).
First monthly exam Gas dynamics Third stage Q1/Water at 15° C flow through a 300 mm diameter riveted steel pipe, E-3 mm with a head loss of 6 m in 300 m length. Determine the flow rate in pipe. Use moody chart. Q2/ Assume a car's exhaust system can be approximated as 14 ft long and 0.125 ft-diameter cast-iron pipe ( = 0.00085 ft) with the equivalent of (6) regular 90° flanged elbows (KL = 0.3) and a muffler. The muffler acts as a resistor with a loss coefficient of KL= 8.5. Determine the pressure at the beginning of the exhaust system (pl) if the flowrate is 0.10 cfs, and the exhaust has the same properties as air.(p = 1.74 × 10-3 slug/ft³, u= 4.7 x 10-7 lb.s/ft²) Use moody chart (1) MIDAS Kel=0.3 Q3/Liquid ammonia at -20°C is flowing through a 30 m long section of a 5 mm diameter copper tube(e = 1.5 × 10-6 m) at a rate of 0.15 kg/s. Determine the pressure drop and the head losses. .μ= 2.36 × 10-4 kg/m.s)p = 665.1 kg/m³
Knowledge Booster
Background pattern image
Mechanical Engineering
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, mechanical-engineering and related others by exploring similar questions and additional content below.
Similar questions
SEE MORE QUESTIONS
Recommended textbooks for you
Text book image
Elements Of Electromagnetics
Mechanical Engineering
ISBN:9780190698614
Author:Sadiku, Matthew N. O.
Publisher:Oxford University Press
Text book image
Mechanics of Materials (10th Edition)
Mechanical Engineering
ISBN:9780134319650
Author:Russell C. Hibbeler
Publisher:PEARSON
Text book image
Thermodynamics: An Engineering Approach
Mechanical Engineering
ISBN:9781259822674
Author:Yunus A. Cengel Dr., Michael A. Boles
Publisher:McGraw-Hill Education
Text book image
Control Systems Engineering
Mechanical Engineering
ISBN:9781118170519
Author:Norman S. Nise
Publisher:WILEY
Text book image
Mechanics of Materials (MindTap Course List)
Mechanical Engineering
ISBN:9781337093347
Author:Barry J. Goodno, James M. Gere
Publisher:Cengage Learning
Text book image
Engineering Mechanics: Statics
Mechanical Engineering
ISBN:9781118807330
Author:James L. Meriam, L. G. Kraige, J. N. Bolton
Publisher:WILEY
How to balance a see saw using moments example problem; Author: Engineer4Free;https://www.youtube.com/watch?v=d7tX37j-iHU;License: Standard Youtube License