Calculus-Sec-11-3-Solution


 

    11.3       The Scalar or Dot Product                  by SGLee - HSKim - JHLee

 

 

1. Determine the dot product of two vectors if their lengths are 8 and  and the angle between them is .

 

      Let the vectors be  and .

     Then,  by definition of the dot product.

 




2-6. Find the dot product.

 

 2. 

           http://matrix.skku.ac.kr/cal-lab/cal-11-3-2.html

 

     .

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

     Open resources in http://math1.skku.ac.kr/pub/ )




3. 

 

 

     .




4. 

 

     .




5. 

 

    .




6. , and the angle between  and  is .

 

     .




7-9. Compute the angle between the vectors.

 

7. 

 

 

     and .

     From the definition of the dot product, we have .

     Hence the angle between  and  is .

     That is,  and  are orthogonal.




 8. (a) 

           (b) 

           http://matrix.skku.ac.kr/cal-lab/cal-11-3-8.html

 

 




(a)




(b)




9. 

 

 

     , and

     .

     From the definition of the dot product, we have  and

      .




 10. Verify whether the given vectors are orthogonal, parallel, or neither.

            (a) 

 

 

     Since  and  are orthogonal.

 




(b)  

      http://matrix.skku.ac.kr/cal-lab/cal-11-3-8.html

 

      Parallel.




 (c) 

     Parallel.




 11. Determine  such that the vectors  and  are orthogonal.

            http://matrix.skku.ac.kr/cal-lab/cal-11-3-11.html

 

 




12. Find a unit vector that is orthogonal to both  and .

 

     .







13-14. Find the direction cosines and direction angles of the vector.

          (Give the direction angles correct to the nearest degree.)

 

13. 

 

 

      Since , the direction cosines of the vector are

      .

      Hence, .




14. 

 

     direction cosine: ,

     direction angle: .




15.Prove that the vector  known as orthogonal projection of , is orthogonal to .

 

 

     

      This proves the result.

 




16-19. Find the scalar and vector projections of  onto  and orthogonal projection of , and .

 

16. 

 

      scalar projection: ,

      vector projection: ,

      orthogonal projection: .




 17. 

            http://matrix.skku.ac.kr/cal-lab/cal-11-3-17.html 

 

 













18. 

 

 

      scalar projection: ,

      vector projection: ,

      orthogonal projection: .




19. 

 

 

     , so  and

      .

      And .




20. Prove that the distance from a point  to the line  is .

     Hence, find the distance from the point  to the line .




21. Prove the Cauchy-Schwarz Inequality: .

 

      Since ,

      .

22. Prove the Triangle Inequality: .

 

 

        Note that it is enough to prove .

        Consider the  L · H · S ,

            

                                          by (21)

                                        

       Hence we have

                        

                             

23. Prove the Parallelogram Law: .

 

      and

     .

      Adding these two equations gives .

24. Show that vectors a and b are orthogonal if and only if .

25.  Show that  if and only if  is orthogonal to .

26. Give geometric interpretation of the above two exercises.

 

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