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.

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 .

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 .

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. .

13.

and , so a vector orthogonal to the plane through  and  is

.

That is,  is orthogonal to the plane through  and .

14-15. Find the area of triangle .

14. .

15.

.

16-17. Find the volume of the parallelepiped with adjacent edges , and .

16.

17.

and .

.

So, the volume of the parallelepiped is  cubic units.

18. Show that the vectors  , and  are not coplanar.

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 .