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
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
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
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)
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?
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.