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.

.

(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)

(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)

Parallel.

(c)

Parallel.

11. Determine  such that the vectors  and  are orthogonal.

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: .

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.