Calculus-Sec-11-4-Solution


   11.4     The Vector or Cross Product          by SGLee - HSKim - JHLee

 

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 

 

 




2. 

 

      

      Now,  and

      

      So,  is orthogonal to both  and .




3. 

 

      .




4. 

 

      .




5. 

 

      .




6. If  and , find  and .

 

 

      .




7. If , and , show that .

 

      (i)

                         <-- (1, -1, 1)

                      <-- (4, 8, 4)

      (ii)

          

          

         Hence, .




 8. Find two unit vectors orthogonal to both  and .

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

 

 

     

      Thus, two unit vectors orthogonal to both are ,

      that is,  and  .




9. Find two unit vectors orthogonal to both  and .

 

      .

      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 

 

      We may think of these points in plane in the space.

                                 




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

 

 

      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 

 

 




13. 

 

 

       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 

 

 




15. 

 

      .




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 

 

 




17. 

 

 

      and .

     .

      So, the volume of the parallelepiped is  cubic units.




 

 18. Show that the vectors  , and  are not coplanar.

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

 

       Not coplanar.




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

 

 

      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 

 

 

      False.

      If , then , hence  is perpendicular to .

      This can happen if .

      For example, let  and , then .

 

(b) If  then 

 

  

       False.

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

 

(c) If  and  then 

 

      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 .

                                            

 

                                                 Back to Part II