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 
.
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 
.
.
14. 
, 
, 
.