College of Engineering Electronics and Communication Problem Set in Vector Analysis

June 16, 2018 | Author: Nicole Kyna | Category: Documents


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Republic of the PhilippinesLaguna State Polytechnic UniversityProvince of LagunaRepublic of the PhilippinesLaguna State Polytechnic UniversityProvince of Laguna
Republic of the Philippines
Laguna State Polytechnic University
Province of Laguna

Republic of the Philippines
Laguna State Polytechnic University
Province of Laguna




College of EngineeringElectronics and Communication Engineering DepartmentCollege of EngineeringElectronics and Communication Engineering Department
College of Engineering
Electronics and Communication Engineering Department

College of Engineering
Electronics and Communication Engineering Department






Problem Set inVector AnalysisProblem Set inVector Analysis
Problem Set in
Vector Analysis
Problem Set in
Vector Analysis



Paulmino, Ian Paul V.ECE III - ASeptember 26, 2017Paulmino, Ian Paul V.ECE III - ASeptember 26, 2017
Paulmino, Ian Paul V.
ECE III - A
September 26, 2017

Paulmino, Ian Paul V.
ECE III - A
September 26, 2017








ENGR. JONNEL K. PABICO, PECE
Instructor
Vector Analysis Vector Analysis
Vector Analysis
Vector Analysis

1. Specify a unit vector extending from the origin to point A (2, -2, 1)
A=2ax-2ay+az
/A/ = 22+-22+12
aA=2ax3-2ay3-az3
=0.667ax-0.667ay+0.333az


2. Determine the unit vector that extends from the origin to the point B (-2, 3, -5)
A=-2ax+3ay-5az
/A/ =(-2)2+(3)2+(-5)2
= 4+9+25
= 38
=6.164

aA= -2ax6.164+3ay6.164-5az6.164
= 0.324ax+0.487ay-0.811az


3. Find the vector A directed from (2, -4, 1) to (0, -2, 0). Determine also the unit vector A.
A=-2ax+2ay-az
/A/ =(-2)2+(2)2+(1)2
=9
=3

aA= -2ax3+2ay3-az3
= -0.667ax+0.667ay-0.333az



4. For two points P (1, 4, 3) and Q (4, 1, 2) express PQ.
PQ=4-1ax+1-4ay+2-3az
PQ=3ax-3ay-az


Vector AnalysisVector Analysis
Vector Analysis
Vector Analysis

5. For 3 points A (0, -1, 4), B (2, 4, 1) and C (3, 0, 2) express 12(AB)+13(BC).
AB= 2-0ax+4+1ay+1-4az
= 2ax+5ay-3az
BC= 3-2ax+0-4ay+(2-1)az
= ax-4ay-az
AC= 3-0ax+0+1ay+2-4az
= 3ax+ay-2az

=122ax+5ay-3az+13ax-4ay+az+163ax+ay-2az
=ax+52ay-32az+9x3-43ay+az3+12ax+16ay-13az
= 116ax+43ay+32ay


6. A drone travel making a vector of A=3ax+2ay-3az then it travels again making a v vector of B=4bx+2by-3bz. What is the vector if it travels back to the origin in a s t r straight path?
A+B=3+4ax+2+2ay+-3+-3az
=-7ax+4ay-6az
=-7ax-4ay+6az

7. Given the vectors M=-10ax+4ay-8az and N=8ax+7ay-2az find (a) a unit v e vector in the direction of -M+2n , (b) the magnitude of 5ax+N-3M; (c) (c) M2NM+N.

A.) -M+2N=(10ax-4ay+8az) B.) 5ax+N-3M=5ax+8ax+7ay-2az
+16ax+14ay-4az -3ax+12ay-24az =26ax+10ay+4az =43ax-5ay+22az

A=262+102+42 A=432+(-5)2+222
=28.142 =48.559

AA=26ax28.142+10ay28.142+4az28.142
=0.924ax+0.355ay+0.142az
Vector Analysis Vector Analysis
Vector Analysis
Vector Analysis

C.) M2NM+N=M=-102+42+-82 2N=82+72+-22
=13.416 =21.63

M+N=-10ax+4ay-8az+8az+2az
= -2ax+11ay-10az
13.41621.633-2ax+11ay-10az=-580.457ax+3192.512ay+(-2902.283az)

8.) Given three points A4,3,2,B-2,0,5 and C7-2,1: (a) Specify the vector A ex t e n extending from the origin to the point A; (b) give a unit vector extending from the origin tow toward the midpoint of line AB; (c) calculate the length of the perimeter of triangle ABC.

A.) A4,3,2 A=4ax+3ay+2az
B.) M=112A+B Unit Vector
=1124ax+3ay+2az-2ax+5aZ =1ax3.937+1.5ay3.937+3.5ax3.937
=1122ax+7az=ax+15ay+3.5az =0.254ax+0.381ay+0.889az
M=12+1.52+3.52 =3.937
C.) Length of the perimeter of ABC
AB= -6ax-3ay+3az=-62+-32+32=7.348
BC= 9ax-2ay-4az=92+-22+-42=10.05
AC= 3ax-5ay-az=32+-52+12=5.916
AB+BC+AC=23.314


9.) The vector from the origin to the point A is given as (6, -2, 4) and the unit vector dire cte directed from the origin toward point B 2, -2, 113. If points A and B are ten units apa rt apart, find the coordinates of point B.
A=6ax-2ay-4az
B=23,23,13
B-A=10 or
6-238ax-2-238ay-4+138az=10
36-88+4-83B+49B2+16+83B2+19B2=100
B2-8B+56=100 B=2311.75ax-2311.75ay+1311.75az
B2-8B-44=0 = 7.833ax-7.833ay+3.917az
B=8±64-1762=11.75
Vector AnalysisVector Analysis
Vector Analysis
Vector Analysis

10.) Given points A 8,-5,4 find: (a) the distance from A to B, (b) a unit vector directed toward B;
(c) a unit vector directed from the origin toward the midpoint of line AB; (d) the coor- fdd dinates of the point on the line connecting A to B at which the line intersects the plan e plane 2=3.

A. Distance from A to B B. Unit vector directed from A to B
B-A=-10ax+8ay-2az B-AB-A=-10ax12.961+8ay12.961-2az12.961
= 102+82+-22 =-772ax+0.617ay-0.154az
=12.961
C. Unit vector directed from the origin to the midpoint of line AB.
A+B2A+B2=6ax-2ay+6az28.71812
=3ax-ay+3az4.359
=3ax4.359-ay4.359+3az4.359
=0.688ax-0.229ay+0.688az
D. The midpoint (3,1) as determined from apart c happens to have 2 coordinate of 3. This This is the point we are looking for.


11.) A vector field is specified as G=24xyax+12x2+2ay+18z2az. Given two points points P=1,2,-1 and Q=-2,1,3 find: (a) G at P; (b) a unit vector in the direct ion direction of G at Q; (c) a unit vector directed from Q toward P; (d) the equation of the
surface on which G=60.

A. G at P: G(1,2,-1) = (48,36,18) or 48ax+36ay+18az
B. A unit vector in the direction of G at Q-G (-2,1,3) = -48ax+72ay+16az
Ag=48ax183.663+72ay183.663+162az183.663
=0.261ax+0.392ay+0.882az
C. P-QP-Q=3,1,426=3ax26+Gy26-4az26=0.588ax+0.196ay-0.784az
D. 60 =124xyax+12x2+2ay+18z2az6= 10=14xyax+2x2+2ay+3z2az2
=100=16x2y2+4x4+16x2+16+9z4


12.) Given 3 points A (4,3,2), B (-2,0,5), C (7,-2,1)
A. Specify the vector A extending from origin to the point A
=4ax+3ay+2az
Vector Analysis Vector Analysis
Vector Analysis
Vector Analysis

B. Give the unit vector extending from the origin to the midpoint of AB
= 12A+B=124ax+3ay+2az+-2ax+5az
=122ax+3ay+7az =12+1.52+3.52
=ax+1.5ay+3.5az =3.937
A A=1ax3.937+1.5ay3.937+3.5az3.937
=0.254ax+0.381ay+0.889az
C. Calculate the perimeter of ABC
AB=(-6,-3,3) AB+BC+AC
BC=(9,-3,-4) =7.348+10.05+5.916
AC=(3,-5,-1) =23.314


13.) Determine 2F1-2F2=2F1-F2
26j+3j-k-24i-5j+8k=26i+3j-k-4i-5j+8k
12i+6j-2k-8i+10j-16k=22i+8j-9k
4i+16j-18k 4i+16j-18k

14.) For two points P (2, 4, 6) and Q (5, 2, 3) express PQ.
PQ=5-2ax+2-4ay+3-6az
PQ=3ax-2ay-3az


15.) If A & B are (3,4,5) and (6,8,9), find AB
B-A=6-3i+8-4j+9-5k AB=32+42+42
B-A=3i+4j+4k AB=41


16.) If A & B are (2,4,6) and (8,9,10), find AB
B-A=8-2i+9-4j+10-6k AB=62+52+42
B-A=6i+5j+4k AB=77


17.) If A & B are (5,6,7) and (9,10,11), find AB
B-A=9-5i+10-6j+11-7k AB=42+42+42
B-A=4i+4j+4k AB=48
Vector Analysis Vector Analysis
Vector Analysis
Vector Analysis

18.) Use the definition of dot product to find the interior angles of A & B of the triangle defined by points. A( 1,3,2) B(-2,4,5) C(0,-2,1)

32+12+72=7.681
12+52+32=5.916
RAB AC=cos-119(7.681)(5.916) θ=65.284° (interior angle)

BC=22+62+42=7.438
Bθ=-32+-61+-47=-40
cos-1-407.4837.681=-134.101°
=180-134.101° θ=45.899°


19.) Give the cartesian coordinates of the point C(ρ=4.4, φ=-115°, z=2)

x=ρcosφ y=ρsinφ z=2
x=4.4cos-115 y=4.4sin(-115)
x=-1.86 x=-3.99

Z Z C-1.86, -3.99, 2
Z
Z
-1.86ax-3.99ay+2az


20.) Given points A(8,-5,4) and B(-2,3,2) find the unit vector directed from the the origin toward the midpoint of line AB.

A+B2=8-22,-5+32,4+22
A+B2=3ax-ay+3az
=3ax-ay+3az32+12+32
=3ax-ay+3az4.36
=0.69ax-0.23ay+0.69az
Vector AnalysisVector Analysis
Vector Analysis
Vector Analysis

21.) Four 10nC positive charge are located in the z=0 plane at the corners of a square 8cm o on a side. A fifth 10nC positive charge is located at a point 8cm distant from the other other charges. Calculate the magnitude of the total force on this fifth charge for ϵ=ϵ0.
F=42*924πϵad
=42*(10-8)24π(8.85 x 10-12)(0.08)2
F=4.0 x 10-4N


22.) Point charges of 50nC each are located t A(1,0,0), B(-1.0,0), C(0,1,0) and D(0,-1,0) in free free space. Find the total force on the charge at A.

F=50 x 10-924πϵ0RCARCA3+RDARDA3+RBARBA3
F=50 x 10-924πϵ0122+122+28
ax=21.5axμN


23.) A shift is travelling due to @10kph. What must be the velocity of the second ship hvim having 30° of Nort if it is always due to North of the 1st ship.

Cos60=10x
x=10cos60
x=20kph



24.) A truck is moving North at speed of 70km/hr. The exhaust pipe above the truck cab sends sends smokes that make an angle of 20° Eq 5 behind the track. If the wind is blowing direc direct east, What is the wind speed at the location?

Tan70=70x
x=70tan70
x=25.48kmhr
Vector AnalysisVector Analysis
Vector Analysis
Vector Analysis

25.) Specify the unit vector extending from the origin toward the point G(2,2,-1)

G=2ax-2ay-az G=2ax-2ay-az22+22+12
=2i-2j-k g=23ax-23ay-13az


26.) A) Give the Cartesian coordinates of the pt. C (ρ=4.4, ө=-115°, z=2).
X=ρcosө=4.4cos-115°=-1.859
Y=ρsinө=4.4sin-115°=-3.988
Z=2
B) Give the cylindrical coordinates of pt. D(x=-3.1, y=2.6, z=-3).
ρ=(-3.1)2+(2.6)2=4.046
ө=tan-12.6-3.1=-40°=140
z=-3
C) Determine the distance from pt. C to D.
D=(Xd-Xc)2+(YD-YC)2+(ZD-Zc)2
=(-3.1+1.869)2+(2.6+3.988)2+(-3-2)2
=8.363

Given two pts. C(-3,2,1) & D(r=5,ө=20°,Ø=-70°), find a) the spherical coordinates of c. b) the Cartesian coordinates of D. c) distance from C to D.

r=(-3)2+(2)2+(1)2
=3.742
Ө=cos-113.742
=75.5°
Ø=tan-12-3
=-33.69°=146.31
x=5sin20cos-70=0.585
y=5sin20cos-70=-1.61
z=5cos20=4.7
RDC=0.585+3ax+-1.61-2ay+4.7-1az
=3.585ax-3.61ay+3.7az
=(3.585)2+(-3.61)2+(3.71)2
=6.297
Vector AnalysisVector Analysis
Vector Analysis
Vector Analysis
Transform the vector B=yax-xay+zaz to cylindrical coordinates.
Solution:
Bρ=B·aρ
=(yax-xay)·aρ
=ρsinөax·aρ-ρcosөay·aρ
=ρsinөcosө-ρcosөsinө
=0
Bө=B·aө
=(yax-xay)·aө
=ρsinөax·aө-ρcosөay·aө
=ρsinө-sinө-ρcosөcosө
=-ρsin2ө-ρcos2ө
=-ρ(sin2ө+cos2ө)
=-ρ
Bz=(yax-xay+zaz)·az
=zaz·az
=z
B=-ρaө+zaz

For two points P (4, 6, 8) and Q (7, 4, 5) express PQ.
PQ=7-4ax+4-6ay+5-8az
PQ=3ax-2ay-3az


Given the points M(-1,2,1) N(3,-3,0) P(-2,-3,4) find; A.) RMN B.) RMP C.) RMP

RMN = N-M
=3,-3,0-(-1,2,1)
=4,-5,-1
RMN=4ax-5ay-az


B.) RMP = P-M RMN+RMP
RMP =P-M =4,-5,-1+(-1,-5,-5)
=-2,-3,-4--1,2,1 =(3,-10,-6)
RMP =(-1,-5,-5) =3ax-10ay-6az

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