So, is orthogonal to both and .
6. If and , find and .
7. If , , and , show that .
<-- (1, -1, 1)
<-- (4, 8, 4)
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. , , ,
So, the volume of the parallelepiped is cubic units.
18. Show that the vectors , , and are not coplanar.
19. Determine whether the points , , , and lie in the same plane.
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
If , then , hence is perpendicular to .
This can happen if .
For example, let and , then .
(b) If then
If , then , which implies that is parallel to , which of course can happen if .
(c) If and then
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 .