Kenneth Kuttler

14.8. EXERCISES 307

8. Find the area of the triangle determined by the three points (1,2,3) ,(2,3,4) and(0,1,2) . Did something interesting happen here? What does it mean geometrically?

9. Find the area of the parallelogram determined by the vectors (1,2,3) and (3,−2,1) .

10. Find the area of the parallelogram determined by the vectors (1,−2,2) and (3,1,1) .

11. Find the volume of the parallelepiped determined by the vectors i−7j−5k,i−2j−6k,3i+2j+3k.

12. Find the volume of the parallelepiped determined by the vectors i+j−5k,i+5j−6k,3i+j+3k.

13. Find the volume of the parallelepiped determined by the vectors i+6j+5k,i+5j−6k,3i+j+k.

14. Suppose a,b, and c are three vectors whose components are all integers. Can youconclude the volume of the parallelepiped determined from these three vectors willalways be an integer?

15. What does it mean geometrically if the box product of three vectors gives zero?

16. It is desired to find an equation of a plane parallel to the two vectors a and b contain-ing the point 0. Using Problem 15, show an equation for this plane is∣∣∣∣∣∣

x y za1 a2 a3b1 b2 b3

∣∣∣∣∣∣= 0

That is, the set of all (x,y,z) such that

x∣∣∣∣ a2 a3

b2 b3

∣∣∣∣− y∣∣∣∣ a1 a3

b1 b3

∣∣∣∣+ z∣∣∣∣ a1 a2

b1 b2

∣∣∣∣= 0

17. Using the notion of the box product yielding either plus or minus the volume of theparallelepiped determined by the given three vectors, show that

(a×b) ·c= a· (b×c)

In other words, the dot and the cross can be switched as long as the order of thevectors remains the same. Hint: There are two ways to do this, by the coordinatedescription of the dot and cross product and by geometric reasoning.

18. Is a×(b×c) = (a×b)×c? What is the meaning of a×b×c? Explain. Hint: Try(i×j)×j.

19. Verify directly that the coordinate description of the cross product a×b has theproperty that it is perpendicular to both a and b. Then show by direct computationthat this coordinate description satisfies

|a×b|2 = |a|2 |b|2 − (a ·b)2 = |a|2 |b|2(1− cos2 (θ)

)where θ is the angle included between the two vectors. Explain why |a×b| has thecorrect magnitude. All that is missing is the material about the right hand rule. Verify

14.8.10.11.12.13.14.15.16.17.18.19.EXERCISES 307. Find the area of the triangle determined by the three points (1,2,3),(2,3,4) and(0, 1,2). Did something interesting happen here? What does it mean geometrically?Find the area of the parallelogram determined by the vectors (1,2,3) and (3,—2,1).Find the area of the parallelogram determined by the vectors (1,—2,2) and (3, 1,1).Find the volume of the parallelepiped determined by the vectors 1-77 —5k,t—2]7 —6k,34+ 27 + 3k.Find the volume of the parallelepiped determined by the vectors 2+ 7 —5k,t+5j —6k,34+9+3k.Find the volume of the parallelepiped determined by the vectors 2+ 67 +5k,2+5]7 —6k,3i+9 +k.Suppose a,b, and c are three vectors whose components are all integers. Can youconclude the volume of the parallelepiped determined from these three vectors willalways be an integer?What does it mean geometrically if the box product of three vectors gives zero?It is desired to find an equation of a plane parallel to the two vectors a and b contain-ing the point 0. Using Problem 15, show an equation for this plane isx y 2aq, a @&4 |= 0by bo bsThat is, the set of all (x,y,z) such thata2 43br bya, a3by bgay ay |_x —y +2] $ by =0Using the notion of the box product yielding either plus or minus the volume of theparallelepiped determined by the given three vectors, show that(a x b)-c=a-(bxc)In other words, the dot and the cross can be switched as long as the order of thevectors remains the same. Hint: There are two ways to do this, by the coordinatedescription of the dot and cross product and by geometric reasoning.Is ax (b x c) = (a x b) x c? What is the meaning of a x b x c? Explain. Hint: Try(tx j) xj.Verify directly that the coordinate description of the cross product a x b has theproperty that it is perpendicular to both a and b. Then show by direct computationthat this coordinate description satisfiesJa x b|? = |al? |b)? — (a-b)* = |a|? |b)? (1 cos? (8)where @ is the angle included between the two vectors. Explain why |a@ x b| has thecorrect magnitude. All that is missing is the material about the right hand rule. Verify