11.4 The Vector or Cross Product

1-5. Find the cross product and verify that it is orthogonal to both and .

(You may do it with Sage in http://math1.skku.ac.kr/ . Open resources in http://math1.skku.ac.kr/pub/)

 

1. , .

http://matrix.skku.ac.kr/cal-lab/cal-11-4-1.html

 Sol)

 

a=vector(QQ, [1, -1,1])

b=vector(QQ, [2,0,3]) 

c=a.cross_product(b)  

cn=c.norm()

e=c/cn    

show(c)

show(e)

show(e.dot_product(a))

show(e.dot_product(b))

  (−3,−1,2)

  (−3/14*sqrt(14),−1/14*sqrt(14),7/1*sqrt(14))

  0

  0


2. , .

 Sol) .


3. , .

 Sol)

      Now, and

      

      So, is orthogonal to both and .


4. .

 Sol) .


5. .

 Sol) .


6. If and , find and .

 Sol) , .


7. If , , and , show that .

 Sol) (i)

          

      (ii)

          

      Hence, .


8. Find two unit vectors orthogonal to both and .

http://matrix.skku.ac.kr/cal-lab/cal-11-4-8.html

 Sol)

 

a=vector(QQ, [1, -1,2])

b=vector(QQ, [3,0,1]) 

c=a.cross_product(b)  

cn=c.norm()

e=c/cn    

show(e)

show(-e) 

  (−1/35*sqrt(35),1/7*sqrt(35),3/35*sqrt(35))

  (1/35*sqrt(35),−1/7*sqrt(35),−3/35*sqrt(35))


9. Find two unit vectors orthogonal to both and .

 Sol) .

      Thus, two unit vectors orthogonal to both are , that is, and

      .


10. Find the area of the parallelogram with vertices , , and .

http://matrix.skku.ac.kr/cal-lab/cal-11-4-10.html

 Sol) .

 

A=(0,1)

C=(2,1)

B=(1,4)

D=(1,-2)

show(point([A,B,C,D])+line([A,B])+line([B,C])+line([C,D])+line([D,A]))

 

 

11.Find the area of the parallelogram with vertices , , , and .

 Sol) The parallelogram is determined by the vectors and , so the area of parallelogram is

      


12-13. Find a vector perpendicular to the plane through the points , , and .


12. , , .

http://matrix.skku.ac.kr/cal-lab/cal-11-4-12.html

 Sol)

 

P=vector([1,0,0]);

Q=vector([4,1,-1]);

R=vector([2,-1,-2]);

PQ=Q-P;

PR=R-P;

PQ.cross_product(PR)

  (-3, 5, -4)


13. , , .

 Sol) and , so a vector orthogonal to the plane through and is

      .

      That is, is orthogonal to the plane through and .


14-15. Find the area of triangle .

 

14. , , .

http://matrix.skku.ac.kr/cal-lab/cal-11-4-14.html

 Sol)

 

P=vector([1,0,0]);

Q=vector([4,1,-1]);

R=vector([2,-1,-2]);

PQ=Q-P;

PR=R-P;

CP=PQ.cross_product(PR);

CP.norm()

  5*sqrt(2)

  

15. , , .

 Sol) .


16-17. Find the volume of the parallelepiped with adjacent edges , , and .


16. , , , .

http://matrix.skku.ac.kr/cal-lab/cal-11-4-16.html

 Sol)

 

P=vector([2,0,1]);

Q=vector([4,2,0]);

R=vector([3,1,-1]);

S=vector([1,1,0]);

PQ=Q-P;

PR=R-P;

PS=S-P;

CP=PQ.cross_product(PR);

PS.inner_product(CP).abs()

  6


17. , , , .

 Sol) and .

     .

      So, the volume of the parallelepiped is cubic units.

18.Show that the vectors , , and are coplanar.

http://matrix.skku.ac.kr/cal-lab/cal-11-4-18.html

 Sol) Not coplanar.

 

a=vector(QQ, [1, 1, 0])

b=vector(QQ, [2, -1, 4]) 

c=vector(QQ, [2, 1, 4])

M=matrix(QQ, [a,b,c]);M

print M

print M.det()

  [ 1  1  0]

  [ 2 -1  4]

  [ 2  1  4]

  -8


19.Determine whether the points , , , and lie in the same plane.

 Sol) and .

     .

      Thus, the volume of the parallelepiped determined by and is . This says that these vectors lie in the same plane. Therefore, their initial and terminal points and also lie in the same plane.

 

20. A wrench 40cm long lies along the positive -axis and grips a bolt at the origin. A force is applied in the direction at the end of the wrench. Find the magnitude of the force needed to supply of torque to the bolt.


21.Suppose that . Prove or disprove the following statements.

 (a) If then

 Sol) False.

     If , then , hence is perpendicular to .

     This can happen if .

     For example, let and , then .


 (b) If then

  Sol) False.

      If , then , which implies that is parallel to , which of course can happen if .


 (c) If and then

 Sol) True.

     Since , is perpendicular to , by part (a). From part (b), is parallel to . Since , and is both parallel and perpendicular to , we have . Hence .


22. Show that .


23. If and , then find .