11.4 The Vector or Cross Product

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. , .

Sol)

 a=vector(QQ, [1, -1,1]) b=vector(QQ, [2,0,3])  c=a.cross_product(b)   cn=c.norm() e=c/cn     show(c) show(e) show(e.dot_product(a)) show(e.dot_product(b))

(−3,−1,2)

(−3/14*sqrt(14),−1/14*sqrt(14),7/1*sqrt(14))

0

0

2. , .

Sol) .

3. , .

Sol)

Now, and

So, is orthogonal to both and .

4. .

Sol) .

5. .

Sol) .

6. If and , find and .

Sol) , .

7. If , , and , show that .

Sol) (i)

(ii)

Hence, .

8. Find two unit vectors orthogonal to both and .

Sol)

 a=vector(QQ, [1, -1,2]) b=vector(QQ, [3,0,1])  c=a.cross_product(b)   cn=c.norm() e=c/cn     show(e) show(-e)

(−1/35*sqrt(35),1/7*sqrt(35),3/35*sqrt(35))

(1/35*sqrt(35),−1/7*sqrt(35),−3/35*sqrt(35))

9. Find two unit vectors orthogonal to both and .

Sol) .

Thus, two unit vectors orthogonal to both are , that is, and

.

10. Find the area of the parallelogram with vertices , , and .

Sol) .

 A=(0,1) C=(2,1) B=(1,4) D=(1,-2) show(point([A,B,C,D])+line([A,B])+line([B,C])+line([C,D])+line([D,A]))

11.Find the area of the parallelogram with vertices , , , and .

Sol) 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. , , .

Sol)

 P=vector([1,0,0]); Q=vector([4,1,-1]); R=vector([2,-1,-2]); PQ=Q-P; PR=R-P; PQ.cross_product(PR)

(-3, 5, -4)

13. , , .

Sol) 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. , , .

Sol)

 P=vector([1,0,0]); Q=vector([4,1,-1]); R=vector([2,-1,-2]); PQ=Q-P; PR=R-P; CP=PQ.cross_product(PR); CP.norm()

5*sqrt(2)

15. , , .

Sol) .

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

16. , , , .

Sol)

 P=vector([2,0,1]); Q=vector([4,2,0]); R=vector([3,1,-1]); S=vector([1,1,0]); PQ=Q-P; PR=R-P; PS=S-P; CP=PQ.cross_product(PR); PS.inner_product(CP).abs()

6

17. , , , .

Sol) and .

.

So, the volume of the parallelepiped is cubic units.

18.Show that the vectors , , and are coplanar.

Sol) Not coplanar.

 a=vector(QQ, [1, 1, 0]) b=vector(QQ, [2, -1, 4])  c=vector(QQ, [2, 1, 4]) M=matrix(QQ, [a,b,c]);M print M print M.det()

[ 1  1  0]

[ 2 -1  4]

[ 2  1  4]

-8

19.Determine whether the points , , , and lie in the same plane.

Sol) 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

Sol) False.

If , then , hence is perpendicular to .

This can happen if .

For example, let and , then .

(b) If then

Sol) False.

If , then , which implies that is parallel to , which of course can happen if .

(c) If and then

Sol) 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 .