11.3 The Scalar or Dot Product

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

 Sol) 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

Sol) .


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

b=vector([4,6]);

a.inner_product(b); 

  2


3. , .

 Sol) .


4. , .

 Sol) .


5. , .

 Sol) .


6. , and the angle between and is .

 Sol) .


7-9. Compute the angle between the vectors.


7. , .

 Sol) , and .

      From the definition of the dot product, we have .

      Hence the angle between and is .


8. ,

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

 Sol)

 

var('t')

a=vector([6,-3,4]);

b=vector([2,0,-3]);

solve(a.inner_product(b)/(a.norm()*b.norm())==cos(t),t)

  [t == 1/2*pi]


9. , .

 Sol) , , and

     .

     From the definition of the dot product, we have and

     .


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

 (a) , .

 Sol) orthogonal.


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

  , .

   Sol) parallel.

 

var('t')

a=vector([6,3]);

b=vector([4,2]);

solve(a.inner_product(b)/(a.norm()*b.norm())==cos(t),t)

  [t == 0]


 (c) ,

 Sol) Parallel.


11. Determine such that the vectors and are orthogonal.

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

 Sol)

 

var('t')

a=vector([2, -6, t])

b=vector([t, t, t^2])

x=a.inner_product(b)

solve([x==0], t)

  [t == -2, t == 2, t == 0]


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

 Sol) .


13-14. Find the direction cosines and direction angles of the vector. (Give the direction angles correct to the nearest degree.)


13. .

 Sol) Since , the direction cosines of the vector are

      .

      Hence, .


14. .

 Sol) direction cosine: ,

     direction angle: , , .


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

 Sol)

      This proves the result.

 

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


16. , .

 Sol) scalar projection: ,

     vector projection: ,

     orthogonal projection: .


17. , .

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

 Sol)

 

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

b=vector(QQ, [4, 3, 3])

ab=a.inner_product(b)  

an=a.norm()

scal_proj=ab/an

scal_proj 

  -1/3


 

vec_proj=scal_proj/an*a

vec_proj 

  (-2/9, 1/9, 2/9)


 

orth_proj=b-vec_proj

orth_proj 

  (38/9, 26/9, 25/9)


 

vec_proj.inner_product(orth_proj)

  0


18. , .

 Sol) scalar projection: ,

     vector projection: ,

     orthogonal projection: .


19. , .

 Sol) , 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: .

 Sol) Since , .


22. Prove the Triangle Inequality: .


23. Prove the Parallelogram Law: .

 Sol) and

     .

     Adding these two equations gives .