11.2 Vectors


1-2. Determine .


1. , .

 Sol) .


2. , .

 Sol) .


3-4. Find the sum of the given vectors.


3. , .

 Sol) .


4. , .

http://matrix.skku.ac.kr/cal-lab/cal-11-2-4.html

 Sol)


v1=vector([0,1,-4]);

v2=vector([0,2,0]);

v1+v2

  (0, 3, -4)


5-8. Compute and .


5. , .

 Sol) ,

     ,

      ,

      .


6. , .

http://matrix.skku.ac.kr/cal-lab/cal-11-2-6.html

 Sol)

 

a=vector([5,-1,3])

b=vector([-1,3,-2])

a.norm() 

  sqrt(35)


 

a+b

    (4, 2, 1)

 

2*a-3*b 

  (13, -11, 12)


 

(a-b).norm() 

  sqrt(77)


7. .

 Sol) ,

     ,

      ,

      .


8. .

http://matrix.skku.ac.kr/cal-lab/cal-11-2-8.html

 Sol)

 

a=vector([3,-4,0]);

b=vector([1,-1,1]);

print a.norm()

print a+b

print 2*a-3*b

print (a-b).norm()

  5

  (4, -5, 1)

  (3, -5, -3)

  sqrt(14)


9. Determine a unit vector that has the same direction as .

 Sol) The vector has length , so the unit vector with the same direction is .


10. Find a vector that has the same direction as but has length 6.

http://matrix.skku.ac.kr/cal-lab/cal-11-2-10.html

 Sol)

 

v1=vector([2,-4,-2]);

v2=6/v1.norm()*vector([2,-4,-2]);

print v2

v2.norm()

  (sqrt(6), -2*sqrt(6), -sqrt(6))

  6


11. A clothesline is tied between two poles, 6m apart. The line is quite taut and has negligible sag. When a wet shirt with a mass of 0.8kg is hung at the middle of the line, the midpoint is pulled down 6cm. Find the tension in each half of the clothesline.

 Sol) Let and represent the tension vectors in each side of the clothe line as shown in the figure. Then and have equal vertical components and opposite horizontal components, so and . By similar its of triangles, .

      The force due to gravity acting on the shirt has magnitude , hence we have . The resultant of the tensile forces counter balances , so

      

      and .

      Thus, the tensions are .


 

12.The tension at each end of the chain has magnitude 50N. What is the weight of the chain?


 13. (a) Draw the vectors , , and .

http://matrix.skku.ac.kr/cal-lab/cal-11-2-13.html

 Sol)

 

z=(0,0)

a=(2,-3)

b=(-2,-1)

c=(7,-5)

A=arrow(z,a)+arrow(z,b)+arrow(z,c)

A=A+text("a",(a[0]+0.3,a[1]+0.3))+text("b",(b[0]+0.3,b[1]+0.3))+text("c",(c[0]+0.3,c[1]+0.3))

show(A) 


 


    (b) Show, by means of a sketch, that there are scalars and such that .

 Sol)

 

j=2

k=-1

z=(0,0)

a=(2,-3)

b=(-2,-1)

c=(6,-5)

A=arrow(z,a,color=(2,1,1))+arrow(z,b,color=(3,1,0))+arrow(z,c,color='black')

A=A+arrow(z,(j*a[0],j*a[1]),color='blue')+arrow(z,(k*b[0],k*b[1]),color='green')

A=A+arrow((k*b[0],k*b[1]),c,color='red')+arrow((j*a[0],j*a[1]),c,color='red')

A=A+text("a",(a[0]+0.3,a[1]+0.3))+text("ja",(j*a[0]+0.3,j*a[1]+0.3))+text("b",(b[0]+0.3,b[1]+0.3))

A=A+text("kb",(k*b[0]+0.3,k*b[1]+0.3))+text("c",(c[0]+0.3,c[1]+0.3))

A=A+point((a, b, c, (j*a[0],j*a[1]),(k*b[0],k*b[1])), rgbcolor='brown', size=30)

show(A) 


 


    (c) Find the exact values of and .

 Sol)

 

var('j, k')

a=(2,-3)

b=(-2,-1)

c=(6,-5)

Ab = matrix(QQ,2,3,[a[0], b[0],c[0],a[1],b[1],c[1]])

Ab.echelon_form()

  [ 1  0  2]

  [ 0  1 -1]


14. If and , describe the set of all points such that .

 Sol)

     

     

     ∴ the surface of a sphere with a center and a radius .