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.
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.
(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.
(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: 
.
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.
8. (a) 
, 
, 
17. 
, 
